SSM_A2_NLB_SBK_Lab13.inddM_A2_NLB_SBK_Lab13.indd PagePage 638638 6/12/086/12/08 2:42:452:42:45 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_LAB/SM_A2_Lab_13Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_LAB/SM_A2_Lab_13

LAB Graphing a Circle and a Polar Equation 13 Graphing Calculator Lab (Use with Lessons 91, 96)

Graphing a Circle in Function Mode

2 2 Graphing 1. Enter the equation of the circle (x - 3) + (y - 5) = 21 as the two Calculator functions, y = ± √21 - (x - 3)2 + 5. Refer to Calculator Lab 1 a. Press to access the Y= equation on page 19 for graphing a function. editor screen. 2 b. Type √21 - (x - 3) + 5 into Y1 2 and - √21 - (x - 3) + 5 into Y2

c. Press to graph the functions. d. Use the or button to adjust the window. The graph will have a more circular shape using 5 rather than 6.

Graphing a Polar Equation

2. Change the mode of the calculator to polar mode. a. Press .Press two times to highlight RADIAN, and then press enter.

b. Press to FUNC, and then two times to highlight POL, and then press .

c. Press two more times to SEQUENTIAL, press to highlight SIMUL, and then press .

d. To exit the MODE menu, press . 3. Enter the equation of the circle (x - 3)2 + (y - 5)2 = 34 as the polar equation r = 6 cos θ - 10 sin θ.

a. Press to access the r1 = equation editor screen.

b. Type in 6 cos θ - 10 sin θ into r1 by pressing Online Connection www.SaxonMathResources.com .

638 Saxon Algebra 2 SSM_A2_NLB_SBK_Lab13.inddM_A2_NLB_SBK_Lab13.indd PagePage 639639 6/13/086/13/08 3:45:473:45:47 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_LAB/SM_A2_Lab_13Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_LAB/SM_A2_Lab_13

Graphing 4. Adjust the viewing window to polar values. Calculator a. Press and type in the following values: When graphing an θ min = 0 Xmin = -16 Ymin = -15 equation in DEGREE mode, set the θ min θ max = 2π Xmax = 21 Ymax = 10 value to 0, the θ max θ step = π/16 Xscl = 1 Yscl = 1 value to 360, and the θ step to 1 to graph the A similar result may be obtained by pressing 5. entire equation with points at every degree. b. The θ min and θ max values determine the minimum and maximum input for the equation. The θ step value determines the increments of the input.

Graphing 5. Press to graph the equation. Calculator The points on the graph are formed by the θ step values. Then, the points are connected with lines. For curved graphs choose small θ step values so that a smoother curve is Lab Practice produced. a. Graph the equation of the circle (x + 2)2 + (y - 7)2 = 14 in function mode.

b. Graph the equation of the limacon r = 1 + 3 cos θ using the given viewing window. θ min = 0 Xmin = -2 Ymin = -3 θ max = 2π Xmax = 5 Ymax = 3 θ step = π/16 Xscl = 1 Yscl = 1

Lab 13 639 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PPageage 664040 66/10/08/10/08 12:18:4812:18:48 AMAM elhielhi //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091

LESSON Making Graphs and Solving Equations of Circles 91

Warm Up 1. Vocabulary The of a circle is the distance from the center of (63) the circle to any point on the circle. 2. Find the length of the segment with endpoints at (3, 5) and (-10, 7). (41) 3. True or False: The unit circle passes through the point (1, 1). (63)

New Concepts A conic section is a plane figure formed by the intersection of a double right cone and a plane. The circle is one of the four possible conic sections.

Math Language A circle is not a function because it does not pass the vertical line test. If the center of the circle is located at (0, 0), A circle is formed by the intersection of right then, using the Distance Formula, the radius of the circle cone and a plane that is given by r = √ (x - 0)2 + (y - 0)2 . Simplifying the is parallel to the base of radicand and squaring both sides gives r2 = x2 + y2. the cone, except where y the intersection is at the The equation of a circle with center (0, 0) and 8 vertex of the cone. (x, y) radius r is 4 x2 + y2 = r2. r (0, 0) x -8 4 8 The equation of a circle must be transformed into Math Reasoning -4 two functions in order to graph it on a graphing Justify A parabola is a calculator in function mode. Isolate y and then -8 conic section. How can a parabola be formed from enter the positive and negative square roots into a plane and a double- y the equation editor as two functions. The graphs 2 2 right cone? y = √r - x of the two functions are semicircles that, together, form a circle. O x x2 + y2 = r2 y2 = r2 - x2 y = ± √ r2 - x2 y =-√r2 - x 2

Example 1 Graphing Circles Centered at the Origin

a. Sketch the graph of x2 + y2 = 16. y SOLUTION 16 equals the value of the radius

squared, so the radius is 4 units. 2 Step 1: Plot the center at (0, 0). O x -2 2 Step 2: Since the radius is 4, plot the points 4 -2 units above, below, left, and right of the center.

Online Connection Step 3: Sketch a circle that passes through the www.SaxonMathResources.com four points.

640 Saxon Algebra 2 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PagePage 641641 6/4/086/4/08 11:32:2711:32:27 AMAM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

b. Graph x2 + y2 = 10 on a graphing calculator. SOLUTION Solve for y. Graph each of two resulting functions. Graphing Calculator x2 + y2 = 10 2 2 To keep the circle from y = 10 - x looking distorted, press 2 ZOOM and choose y = ± √10 - x ZSquare.

y The center of the circle shown at right is located (x, y) at (h, k). r Using the Distance Formula, the radius of the circle is 4 (h, k) 2 2 given by r = √( x - h) + (y - k) . Squaring both sides x gives r2 = (x - h)2 + (y - k)2. 8

Standard Form of an Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r2. In order to graph a circle, the center and the radius of the circle must be determined. Example 2 Graphing Circles Not Centered at the Origin

Hint a. Sketch the graph of (x + 2)2 + (y - 1)2 = 9. Be careful choosing SOLUTION The radius is √9 , or 3. y the values of h and k, 4 as the formula involves Step 1: Plot the center at (-2, 1). subtraction. Think: Step 2: O x (x + 2)2 = (x – (–2))2. Since the radius is 3, plot the points 3 units above, below, left, and right of the center. -6 -2 2 -2 Step 3: Sketch a circle that passes through these four points. -4

b. Graph (x - 7)2 + (y + 3)2 = 15 on a graphing calculator. SOLUTION Solve for y. Graph each of the two resulting functions. (y + 3)2 = 15 - (x - 7)2 y + 3 = ± √15 - (x - 7)2 y = ± √15 - (x - 7)2 - 3

Lesson 91 641 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PPageage 664242 66/10/08/10/08 12:19:0212:19:02 AMAM elhielhi //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091

Sometimes the center and the radius of a circle are not explicitly given, so it is necessary to use the Distance and/or Midpoint Formula to determine these unknown parts.

For a segment whose endpoints are at (x1, y1) and (x2, y2), Distance Formula Midpoint Formula x + x y + y 2 2 _1 2 _1 2 d = ( x2 - x1) + (y2 - y1) M = , √ ( 2 2 )

Example 3 Writing the Equation of a Circle

a. Write the equation of the circle with center (-3, -1) and radius 7. SOLUTION Substitute h = -3, k = -1, and r = 7 into (x - h)2 + (y - k) 2 = r2. (x - (-3))2 + (y - (-1))2 = 72 (x + 3)2 + (y + 1)2 = 49

b. Write the equation of the circle with center (-2, 4) that contains the point (5, 2). SOLUTION Find the length of the radius by using the Distance Formula. r = √( 5 - (-2))2 + (2 - 4)2 = √7 2 + (-2)2 = √53 Substitute h = -2, k = 4, and r = √53 into (x - h)2 + (y - k)2 = r2. 2 (x - (-2))2 + (y - 4)2 = ( √53 ) (x + 2)2 + (y - 4)2 = 53

c. Write the equation of the circle that has a diameter whose endpoints are located at (3, 1) and (6, 3). SOLUTION The center of the circle is midway between the endpoints of a diameter. Use the Midpoint Formula.

x1 + x2 y1 + y2 3 + 6 1 + 3 M = _ , _ = _ , _ = (4.5, 2) ( 2 2 ) ( 2 2 )

Hint Find the distance between the center and either of the points on the circle. 2 2 The radius will always be r = √( 3 - 4.5) + (1 - 2) greater than 0. A radius less than or equal to 0 = √3.25 indicates an error. Substitute h = 4.5, k = 2, and r = √3.25 into (x - h)2 + (y - k)2 = r2. 2 (x - 4.5)2 + (y - 2)2 = ( √3.25 ) (x - 4.5)2 + (y - 2)2 = 3.25

642 Saxon Algebra 2 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PagePage 643643 6/3/086/3/08 12:16:1812:16:18 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091

Example 4 Application: Astronomy The orbit of Earth around the Sun resembles a circle. Given that Earth travels about 584.3 million miles in one orbit, write an equation for the circle that models Earth’s orbit around the Sun.

Hint SOLUTION The distance Earth travels in one orbit is the circumference of the circle, and C = πd. Let the Sun be at (0, 0) on the coordinate plane, C = πd can also be written as C = 2πr. where a unit represents one million miles. Find the diameter and divide by 2 to find the radius. C= πd 584.3 ≈ πd Substitute 584.3 for C. 185.988 ≈ d Divide both sides by π. Divide the diameter by 2 to find the radius: r ≈ 92.994 ≈ 93. Write the equation of the circle. Substitute r = 93 in x2 + y2 = r2. x2 + y2 = 932 x2 + y2 = 8649 The equation x2 + y2 = 8649 models Earth’s orbit around the Sun, where the Sun is located at (0, 0).

Lesson Practice a. Sketch the graph of x2 + y2 = 36. (Ex 1) b. Graph x2 + y2 = 25 on a graphing calculator. (Ex 1) c. Sketch the graph of (x - 5)2 +(y - 3)2 = 4. (Ex 2) d. Graph (x + 8)2 +(y - 1)2 = 22 on a graphing calculator. (Ex 2) e. Write the equation of the circle with center (6, -7) and radius 9. (Ex 3) f. Write the equation of the circle with center (7, -4) that contains the (Ex 3) point (8, 1). g. Write the equation of the circle that has a diameter whose endpoints are (Ex 3) located at (-2, 3) and (4, 9). h. The orbit of Venus around the Sun resembles a circle. Given that Venus (Ex 4) travels about 421 million miles in one orbit, write an equation for the circle that models Venus’s orbit around the Sun.

Practice Distributed and Integrated

3 _3V 1. Geometry The radius of a sphere is given by r = , where V is the volume of the (70) 4π sphere. Approximate the volume of a sphere with a radius of 4.5 centimeters.

*2. Multi-Step The map of a town is placed on a coordinate grid. A radio station’s (91) transmitter is located at (0, 2), and the signal can be picked up by anyone within 95 miles of the transmitter. Write the equation of the circle that represents the broadcast area for the station’s signal. Show how to use the equation to prove that a person located at (94, 6) can pick up the signal.

Lesson 91 643 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PagePage 644644 6/3/086/3/08 12:16:1912:16:19 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091

3. Given that x2 - 5x + 2 = 0, what needs to be added to both sides of the equation (58) to get a perfect square on the left side?

4. Solve ABC. B (71) 25

A 104° 16 C

5. A spiral staircase has 15 steps. Each step is a sector with a radius of 42 inches and (63) _π a central angle of 8 . What is the length of the arc formed by the outer edge of a step? _3 *6. Error Analysis Two students are finding the period of y = tan x but get different (90) ( 5 ) results. Which student made the mistake?

Student A Student B 2π 2π 10π π π 5π Period = _ = _ = _ Period = _ = _ = _ b 3 3 b 3 3 _ _ 5 5

7. Multi-Step For each value of the leading coefficient a, write the quadratic function (78) in the form y = ax2 + c that has zeros -2 and 2. (Hint: The value of c can be positive or negative.) 1 a. a = 1 b. a = _ c. a = -1 2 d. Graph the functions on the same coordinate system. e. How many quadratic functions exist that have two given zeros?

8. A random number generator will output integers from 1 to 100. What is (55) P(multiple of 5) with the generation of the first number?

9. Find the arc length of a sector with a radius of 8 inches and a central angle (63) of 110°.

*10. Generalize What must be true about the values of h and k for a circle whose center (91) is located in Quadrant II?

11. Explain why f (x) = √3 x + 5 is a radical function. (75) *12. The endpoints of a diameter are located at (-2, -1) and (4, -1). Write an (91) equation for this circle.

13. Optics The focal length of a lens, f, the distance from the lens to the object, o, and (84) _1 = _1 + _1 the distance from the lens to the image, i, are related by the formula f o i . Find f if o is 12 inches and i is 8 inches.

14. Analyze How is deriving a quadratic equation from two identical roots different (83) from deriving a quadratic equation from two unique roots?

644 Saxon Algebra 2 SSM_A2_NLB_SBK_L091.inddM_A2_NLB_SBK_L091.indd PagePage 645645 6/3/086/3/08 12:16:2012:16:20 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L091

15. Sports A sports columnist wants to survey college basketball coaches about whether (73) they think women’s basketball nets should be raised to the same measure as in men’s basketball. The writer mails out surveys to all the coaches and uses only the surveys that are returned. Identify the type of sample described.

*16. Write a quadratic equation whose roots are 7 + 3i and 7 - 3i. (83)

Find the roots of the polynomial function. 17. f(x) = 15x3 + 3x2 - 6x 18. f(x) = (x + 1)(4x2 + 4) - (x + 1)(2x - 3) (76) (76)

*19. Verify Let f(x) = cos(x) and g(x) = tan(x). Compare the graph of f(x) g(x) and (90) h(x) = sin(x).

*20. Identify the period and undefined values of y = tan(12x) + 3. (90) 21. Surveying A surveyor stands on the east end of a boardwalk and finds that the (71) angle formed by the boardwalk and the line of sight with a lighthouse is 76°. The surveyor then walks to the west end and finds the angle formed by the boardwalk and lighthouse to be 60°. Estimate the closest distance from the boardwalk to the lighthouse, given that the boardwalk has a length of 2700 feet.

*22. Graphing Calculator Graph (x - 1)2 + (y + 9)2 = 7 on a graphing calculator. Use the (91) Trace feature to find the y-intercepts.

23. Geography A map of southeast Illinois shows that Flora is 44.4 miles from Carmi (77) and 42.6 miles from Mt. Carmel. Mt. Carmel is 34.4 miles from Carmi. Find the number of square miles enclosed in the triangle that is formed when lines connecting these three cities are drawn. Round to the nearest square mile.

24. College Admissions In 2005, the mean score on the ACT reading test was 21.3 with a (80) standard deviation of 6. The mean score on the mathematics test was 20.7 with a standard deviation of 5. Assume both distributions are normally distributed. Who would you say performed better: a student who got a score of 27 on the reading test or a student who got a score of 24 on the math test? Why?

⎧ 8, if x ≤ 0 25. Evaluate the piecewise function f (x) = ⎨ for x = -0.4, x = 0, (79) ⎩ 3x - 1, if x > 0 and x = 6.

26. Multiple Choice Which is equal to ln e x 2 ? (81) A 2x B x2 C 2 ln x D ln 2 e x

3 *27. Formulate Find the roots of (x3) - a3. (85) 28. Probability The right triangle is inscribed in the circle. Suppose a point is chosen (78) at random inside the circle. What is the probability that the chosen point is NOT inside the triangle? x _9 _5 *29. Write Explain why F = C + 32 and C = (F - 32) describe the same relationship x (88) 5 9 between F and C.

30. P is a point on the terminal side of θ in standard position. Find the exact value of (56) the six trigonometric functions for θ where P(-8, -8).

Lesson 91 645 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PPageage 664646 66/10/08/10/08 12:19:4912:19:49 AMAM elhielhi //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092

LESSON Finding Arithmetic Sequences 92

Warm Up 1. Vocabulary In a function, the (domain range) is made up of (4) / the input values. 2. Solve 18 = 6 + 2 (n - 1). (7) 3. True or False: The answer to a subtraction problem is called the difference. (SB)

New Concepts EExplorationxploration Exploring Sequences In a DVD club, a member receives 8 DVDs the first month and 3 DVDs every month thereafter. Step 1: Complete the table.

Month 12345 Total Number of DVDs 8

Step 2: Describe a pattern in the total number of DVDs. Step 3: Find the total number of DVDs for months 6, 7, and 8. Step 4: Graph the data. What type of function models the data? Why? What is the domain of the function?

A sequence is an ordered list of numbers. Each number in the sequence is a term. A sequence can be finite {2, 4, 6, 8}, or infinite {5, 15, 45, 135, …}. Sequences can be written as functions, where the domain is the set of natural numbers and the range is the set of terms. It is common to use the function name of a for a sequence with the term number, or domain, written as a

subscript. Therefore a1 is the first term of a sequence, a2 is the second term,

and an is the nth term. Math Reasoning If the differences between the consecutive terms in a sequence are the same, then the sequence is an arithmetic sequence. The finite sequence {2, 4, 6, 8} is Justify Show that {5, 15, 45, 135, …} is not arithmetic because the difference between any two consecutive terms is 2. an arithmetic sequence. The difference of 2 is called the common difference. The common difference of any arithmetic sequence is found by subtracting any term from its successive term. It can be used to extend an infinite sequence.

Example 1 Identifying and Using the Common Difference Find the common difference of the arithmetic sequence and use it to find the next three terms. -14, -9, -4, 1, ... SOLUTION Subtract a term from the term that follows it: 1 - (-4) = 5. Find the next three terms by adding 5: 1 + 5 = 6, 6 + 5 = 11, and 11 + 5 = 16. Online Connection www.SaxonMathResources.com The next three terms are 6, 11, and 16.

646 Saxon Algebra 2 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PagePage 647647 6/4/086/4/08 11:47:4311:47:43 AMAM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

Math Language An explicit formula defines the nth term of a sequence in terms of n, meaning that the nth term can be found without knowing the previous term. To find the nth term using a recursive formula, For the arithmetic sequence {10, 14, 18, 22, 26, …}, the previous term must be known. The recursive a1 = 10 + 0(4), a2 = 10 + 1(4), a3 = 10 + 2(4), a4 = 10 + 3(4), formula is a5 = 10 + 4(4), . . . an = 10 + (n - 1)4. an = an-1 + d. Arithmetic Sequence The nth term of an arithmetic sequence is given by

an = a1 + (n - 1)d.

Example 2 Finding the nth Term of an Arithmetic Sequence

a. Find the 15th term of the arithmetic sequence 4, 15, 26, 37, …

SOLUTION Find d: 37 - 26 = 11. Then use the formula an = a1 + (n - 1)d.

a15 = 4 + (15 - 1)11 Substitute 15 for n, 4 for a1, and 11 for d.

a15 = 158 Simplify.

b. Find the 18th term of the arithmetic sequence 11, 5, -1, -7, …

SOLUTION Find d: 5 - 11 = -6. Then use the formula an = a1 + (n - 1)d.

a18 = 11 + (18 - 1)(-6) Substitute 18 for n, 11 for a1, and -6 for d.

a18 = -91 Simplify.

The formula an = am + (n - m)d can be used to find any term when given any two terms of the sequence.

Example 3 Finding the nth Term Given Any Two Terms

a. Find a1 of an arithmetic sequence given that a7 = 34 and a16 = 61.

SOLUTION In the formula an = am + (n - m)d, replace m with 7 and n with 16.

a16 = a7 + (16 - 7)d

61 = 34 + (16 - 7)d Substitute a16 with 61 and a7 with 34. 61 = 34 + 9d Solve for d. 27 = 9d 3 = d

Now find a1. Use the formula an = a1 + (n - 1)d and either of the known terms.

61 = a1 + (16 - 1)3 Substitute 61 for an and 16 for n.

61 = a1 + 45 Solve for a1.

16 = a1 The first term of the sequence is 16.

Check by using a1 to find the 7th term: a7 = 16 + (7 - 1)3 = 16 + 18 = 34.

Lesson 92 647 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PPageage 664848 66/10/08/10/08 12:19:5612:19:56 AMAM elhielhi //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092

b. Find a8 of an arithmetic sequence given that a5 = -180 and a12 = -138.

Math Reasoning SOLUTION In the formula an = am + (n - m)d, replace m with 5 and replace n with 12. Analyze How can you tell that the common a12 = a5 + (12 - 5)d difference will be a positive number before -138 = -180 + (12 - 5)d Substitute a12 with -138 and a5 with -180. doing any calculations? -138 = -180 + 7d Solve for d. 42 = 7d 6 = d

Now find a8. Use an = am + (n - m)d, replacing m with 8, 6 for d, and using one of the given terms.

a12 = a8 + (12 - 8)(6) Use n = 12.

-138 = a8 + (12 - 8)(6) Substitute a12 with -138.

-138 = a8 + 24 Solve for a8.

-162 = a8 The eighth term of the sequence is -162. Since the eighth and twelfth terms are not too far from the fifth term, the answer could be checked by adding the common difference of 6.

a5 a6 a7 a8 a9 a10 a11 a12 -180 -174 -168 -162 -156 -150 -144 -138

Example 4 Application: Salary Richard’s salary is structured so that after the first year, he receives an annual raise of a set amount. What was his starting salary if his salary in his fourth year was $41,135 and his salary in his fourteenth year was $50,785? SOLUTION

Substitute the given information into the formula an = am + (n - m)d. Replace Math Reasoning am with a4, or 41,135, m with 4, an with a14, or 50,785, and n with 14.

Formulate Study what 50,785 = 41,135 + (14 - 4)d is being done to find d and formulate a way to 50,785 = 41,135 + 10d find it without writing an 9650 = 10d equation. 965 = d

Find a1.

50,785 = a1 + (14 - 1)965

50,785 = a1 + 12,545

38,240 = a1

Richard’s starting salary was $38,240.

Check Use a1 to find the 4th term: a4 = 38,240 + (4 - 1)965 = 41,135.

648 Saxon Algebra 2 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PagePage 649649 6/3/086/3/08 12:19:3812:19:38 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092

Lesson Practice _1 _1 a. Find the common difference of the arithmetic sequence 9 2 , 10 4 , 11, (Ex 1) _3 11 4 , …. Then, find the next three terms. b. Find the 44th term of the arithmetic sequence -121, -112, -103, (Ex 2) -94, … c. Find the 33rd term of the arithmetic sequence 0.56, 0.38, 0.2, (Ex 2) 0.02, …

d. Find a1 of an arithmetic sequence given that a3 = 33 and a7 = 63. (Ex 3)

e. Find a3 of an arithmetic sequence given that a7 = 75 (Ex 3) and a12 = -125. f. Beverly’s salary is structured so that after the first year, she receives an (Ex 4) annual raise of a set amount. What was her starting salary if her salary in her sixth year was $31,975 and her salary in her thirteenth year was $36,420?

Practice Distributed and Integrated

*1. Formulate Give an example of a sequence that is formed by adding numbers in a (92) pattern but is not arithmetic.

2. Home Maintenance A homeowner can rake and bag the leaves in his yard in 5 hours. (84) If he works with his children, the job can be done in 3 hours. How long would it take for just the children to rake and bag the leaves in the yard?

*3. Multiple Choice What is the 40th term of the arithmetic sequence whose first term is (92) 13 and whose second term is 15.5? A 97.5 B 100 C 110.5 D 123.5

4. Multi-Step Find the radical function that is the inverse of y = x2 - 6x + 9. Identify (75) its domain and range.

5. Evaluate the inverse trigonometric function cos-1 (-1). Give your answer in both (67) radians and degrees.

6. Write Describe the characteristics of a data set that is normally distributed. (80) *7. Coordinate Geometry If A(5, 2) and B(-4, -3) are the endpoints of a diameter of (91) a circle, what is the equation of the circle?

*8. Write the equation for a circle centered at (2, -9) with a radius of 3. (91) 9. Geometry A geometry teacher has a database with 400 problems about triangles. (73) Can she choose a sample of questions by using a stratified sampling method where the strata are right triangles, isosceles triangles, and scalene triangles? Why or why not?

Lesson 92 649 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PagePage 650650 6/3/086/3/08 12:19:3912:19:39 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092

_v 2 10. Physics A satellite in orbit accelerates according to the equation acceleration = r , (76) for orbital speed v and distance from Earth’s center r. Two satellites are in orbit, one at a distance of r and the other at a distance of 2r. How fast should the outer satellite travel to have the same acceleration as the inner satellite?

11. Multiply (3 + i)(1 - 4i). Write the result in the form a + bi. (69) 12. Multiple Choice Which triangle cannot be solved by using the Law of Cosines? (77) A B 2 4 2 48° 3 3

C D 2 65° 4 2 48° 3

13. Multi-Step Technetium-99m, a radioisotope used to image the skeleton and (81) the heart muscle, has a half-life of about 6 hours. a. How long will it take for 1 mg of technetium-99m to decay to 0.2 mg? (Use the -kt natural decay function N(t) = N0 e .) -kx b. On a calculator, graph the function y1 = 1 · e , using the value of k you found

in part a. Also graph the functions y2 = 0.5 and y3 = 0.2. (suggested window values: x [-1, 20] and y [-0.1, 1.1])

c. Describe what is represented by the relationships among the graphs of y1, y2,

and y3.

14. Delivery Charges A furniture store delivers up to 4 items for a $75 delivery charge. (79) There is an additional charge of $25 per item for each additional item delivered. Write a piecewise function for the cost of having x items delivered.

*15. Find the 17th term of the arithmetic sequence 4, 21, 38, … (92)

Find the roots of the polynomials. 16. y = 4x2 + 5x - 1 17. y = x4 - 16 (58) (76)

18. Geometry A rectangular prism has a volume of 3x3 - 8x2 + 3x + 2. Find the linear (85) dimensions.

*19. Travel A driver on the Pennsylvania turnpike turned on the cruise control at (92) milepost 30, Warrendale, and kept it on until milepost 359, the Delaware River Bridge, exactly 5 hours later. Show how to use an arithmetic sequence to find the speed at which the car traveled during those 5 hours.

*20. Graphing Calculator Graph y = 5 cos 2(x - 3π) using a graphing calculator. (86) Determine its period and phase shift.

2 *21. Verify Show that log5 (125x) equals 8 when x = 5. (87) 22. A principal amount of $1,500 earns interest at an annual rate of 2.7%, compounded (57) monthly. What will its value be after 7 years?

650 Saxon Algebra 2 SSM_A2_NLB_SBK_L092.inddM_A2_NLB_SBK_L092.indd PagePage 651651 6/3/086/3/08 12:19:4012:19:40 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L0Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L092

*23. Write When solving a quadratic inequality, what is meant by the critical values? (89) 24. Solve 6t2 + 5 = 2t2 + 1. Write the solution in terms of i. (62) 25. Analyze Which equations have the same solutions as x2 - 8x + 7 = 0 ? (83) -x2 + 8x - 7 = 0 -x2 + 8x + 7 = 0 -2x2 + 8x - 7 = 0 2x2 - 8x + 14 = 0 2x2 - 16x + 14 = 0

Write the exponential equation in logarithmic form or vice versa. 4 26. 11 = 14,641 27. log9 729 = 3 (64) (64)

*28. Basketball A drafter is creating the plans for a sport and fitness center, which (91) includes an official-size basketball court. On her grid, the units represent feet. The center of the basketball court’s center circle is located at (-14, -1), and another point on the center circle is located at (-14, 2). Write an equation for this circle.

29. Error Analysis A student found that the discriminant of a quadratic equation was 0 (74) and therefore the number of solutions of the equation was also 0. Correct the student’s reasoning.

30. Solve by completing the square: 4x2 = -12x + 4. (78)

Lesson 92 651 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 652652 6/12/086/12/08 3:43:163:43:16 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093

LESSON Solving Exponential Equations 93 and Inequalities

Warm Up 1. Vocabulary In the expression (1.005)x, the is 1.005. (3) 2. What is the base in the expression 240(1.022)x ? (47) 3. What is the base in the expression 50(1 + 0.04)x ? (47)

New Concepts Exponential equations cannot always be solved using the same properties that are used to solve linear equations. One property that can be used to solve simple exponential equations is the Property of Equality.

Math Language Property of Equality for Exponential Functions x y An equation that If a is any positive number other than 1 and a = a , then x = y. contains one or more exponential expressions To apply the property of equality, each side of the equation must be written is an exponential equation. with the same base, a. If they are not written in the same base, they must be written as a power of the same base.

Example 1 Using the Property of Equality

Hint a. Solve for x: 2x = 16 Review Lesson 3 to SOLUTION read about the rules of exponents that are Write 16 as an exponent with a base of 2. necessary to solve 4 exponential equations 16 = 2 like these. Therefore, 2x = 24. By the Property of Equality, x = 4.

b. Solve for c: 9c+2 = 272 SOLUTION Both sides of the equation can be written with a base of 3. 9c+2 = (32)c+2 = 32 c+4 272 = (33)2 = 36 By the Property of Equality, 2c + 4 = 6, so c = 1. 1 c. Solve for t: 2t = _ 16 SOLUTION It may seem like the right side of this equation cannot be written as a power of 2, but recall that it is in fact a negative power of 2. 1 _ = 2-4 16 Online Connection www.SaxonMathResources.com By the Property of Equality, t = -4.

652 Saxon Algebra 2 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 653653 6/10/086/10/08 8:33:488:33:48 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093

Another way to solve an exponential function is to apply the Power Property of Logarithms. Recall from Lesson 72 that the Power Property of Logarithms says that log a p = p log a. To apply the Power Property, take the logarithm of both sides of an exponential equation and simplify the result.

Example 2 Solving by Taking the Logarithm of Each Side

Hint a. Solve for x: 200 = 1.05x Don’t forget the product SOLUTION and quotient rules of logarithms, which 200 = 1.05x may also be useful in x simplifying exponential log 200 = log 1.05 Take the logarithm of both sides. equations as you solve them. log 200 = x log 1.05 Apply the Power Property. log 200 x = _ Divide by log 1.05. log 1.05 x ≈ 108.59 Evaluate.

b. Solve for y: 72y+1 = 4-y SOLUTION 4-y = 72y+1 log 4-y = log72y+1 Take the logarithm of both sides. -y log 4 = (2y + 1)log 7 Apply the Power Property. -y log 4 = 2y log 7 + log 7 Distribute. - y log 4 - 2y log 7 = log 7 Subtract 2y log 7 from each side. y(-log 4 - 2 log 7) = log 7 Factor. log 7 y = __ Division Property of Equality. (-log 4 - 2 log 7) y ≈ -0.37 Evaluate.

If e is used in an exponential equation, solve the equation by taking the natural log of both sides. Recall the inverse property of logarithms: ln ex = x.

Example 3 Solving Exponential Equations with e

a. Solve for x: 5 = 6e x SOLUTION 6ex = 5 ln 6e x = ln 5 Take the logarithm of both sides. ln 6 + ln ex = ln 5 Apply the Product Property. ln 6 + x = ln 5 Apply the Inverse Property. x ≈ -0.18 Subtract and evaluate.

Lesson 93 653 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 654654 6/4/086/4/08 12:28:1912:28:19 PMPM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

Example 4 Solving Exponential Inequalities

a. Suppose you receive a gift of 1 cent on the first day, 2 cents on the second day, and so on, receiving double the amount on each new day. What is the first day on which you will receive more than $1 million? SOLUTION $1 million = $1,000,000 = 100,000,000 cents = 108 cents On day 1 you receive 20 cents, on day 2 you receive 21 cents, and so on, receiving 2n-1 cents on day n. 2n-1 > 108 Write an inequality. log 2n-1 > log 108 Take the logarithm of both sides. log 2n-1 > 8 Simplify. (n - 1)log 2 > 8 Apply the Power Property of Logarithms. 8 n - 1 > _ Divide both sides by log 2. log 2 8 n > _ + 1 ≈ 27.6 Use a calculator to approximate. log 2 The first day on which you will receive more than $1 million is day 28. Check Day 27: 226 cents = 67,108,864 cents = $671,088.64 Day 28: 227 cents = 134,217,728 cents = $1,342,177.28

b. A merchant is having trouble selling a jacket and decides to mark down the price 10% at the beginning of every day. The regular price of the jacket is $199.50. He marks the price down for the first time on day 1. On what day will the price first be below $50?

n n Caution SOLUTION The price on day n is 199.50 (1 - 0.10) , or 199.50(0.90) . n Know the sign of a 199.50(0.90) < 50 Write an inequality. logarithm if you divide n Take the common both sides of an inequality log [199.50(0.90) ] < log 50 by that logarithm. log m is logarithm of both sides. negative if 0 m 1. < < n log m is positive if m > 1. log 199.50 + log (0.90) < log 50 Apply the Product Property of Logarithms. log (0.90)n < log 50 - log 199.50 Subtract log 199.50 from both sides. n log (0.90) < log 50 - log 199.50 Apply the Power Property of Logarithms. log 50 - log 199.50 n > __ Divide both sides by log 0.90 log 0.90 and reverse the inequality symbol. n > 13.13 Use a calculator to approximate. The price will first be below $50 on day 14.

654 Saxon Algebra 2 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 655655 6/2/086/2/08 11:19:3511:19:35 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093

Example 5 Application: Compound Interest How long will it take an investment to double in value at 9% interest compounded quarterly (4 times per year)? _r nt Graphing SOLUTION A = P ( 1 + n) Write the compound interest formula. Calculator Tip 0.09 4t 2P = P 1 + _ Substitute 2P for A because P will The expression ( 4 ) log (2) ÷ 4 log (1.0225) double. Substitute 0.09 for r and 4 for n. is NOT correct for 4t solving Example 4 2 = (1.0225) Divide both sides by P and simplify. log 2 because it equals _ ( 4 ) 4t Take the common logarithm of both sides. (log 1.0225). A correct log 2 = log (1.0225) expression is log 2 = 4t log (1.0225) Apply the Power Property of Logarithms. log (2) ÷ (4 log (1.0225)). log 2 __ = t Divide both sides by 4 log 1.0225. 4 log (1.0225) t ≈ 7.79 years Use a calculator to approximate. The interest is compounded quarterly, so the compounding will likely occur when t has these values: 7, 7.25, 7.50, 7.75, and 8. The value of the investment will not have doubled when t = 7.75, but it will have more than doubled when t = 8. 0.09 (4)(7.75) Check When t = 7.75: A = P 1 + _ = P(1.0225)31 ≈ 1.993P ( 4 ) 0.09 (4)(8) When t = 8: A = P 1 + _ = P(1.0225)32 ≈ 2.038P ( 4 )

Lesson Practice Solve for the variable in each equation. a. 492y = 3432y+3 (Ex 1) b. 3r = 95 (Ex 1) 1 c. 25t = _ (Ex 1) 25 d. 45 = 2.32x (Ex 2) e. 3e2c+1 = 432 (Ex 3) f. Stock in a certain company was purchased on December 31, 2007, for (Ex 4) $1.50 per share. Suppose the price of a share doubles every year. What is the first year in which the price of a share will be greater than $90? g. The regular price of a rug is $325. The price is marked down 6% at the (Ex 4) beginning of each week, beginning with week 1. In what week will the price first be below $200? h. How long will it take an investment to increase 50% in value at 4.8% (Ex 5) interest compounded monthly (12 times per year)?

Lesson 93 655 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 656656 6/2/086/2/08 11:19:3611:19:36 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L093

Practice Distributed and Integrated

1. Fast Food At a fast food restaurant, each take-out order has a ticket in the bottom (73) of the bag asking customers to call an 800 number to register for a prize drawing. Describe the individuals in the population and in the sample.

*2. Multiple Choice Which equation is equivalent to 1.4 = 1.06x? (93) 1.4 log 1.4 A x = 1.4 - 1.06 B x = log 1.4 - log 1.06 C x = _ D x = _ 1.06 log 1.06 3. Gravity The height of a free-falling object (neglecting air resistance) is given by the (78) 2 function h(t) = -4.9t + v0t + h0, where h0 is the initial height in meters, v0 is the initial velocity in meters per second, and h(t) is the height in meters at time t seconds. How long does it take for an object to hit the ground after it is dropped from a height of 14.4 meters?

*4. Verify The first term of an arithmetic sequence is 19 and the common difference (92) is 9. Show that the 13th term is 127.

*5. Geometry A cube has 6 sides. When two cubes are joined together, the figure has (92) 10 sides. When three cubes are joined together, to form a row, there are 14 sides. Show how to use an arithmetic sequence to find the number of sides in a row of 20 cubes joined together.

Determine the domain and range for each of the following. _1 3 2 6. y = -3 x 7. y = (x + 1) + 8 (75) (75)

*8. Find the common difference and use it to find the next three terms. 4, 6.5, 9, … (92) *9. Science Vredefort is an impact crater located in South Africa. Given that its (91) circumference is about 942.5 kilometers, write an equation for the circle that represents the rim of the crater, assuming the center is located at the origin of a coordinate plane.

*10. Solve for x: 43x-6 = 32x+4 (93) _3 _2 11. Generalize Describe two ways to find the solution(s) of + = x - 1 on a (84) 3x + 9 3 graphing calculator.

*12. Graphing Calculator Graph y = tan (2x + 3) on your graphing calculator. Identify (90) its period, undefined values, and phase shift.

*13. Tourist Attractions The world’s largest pair of cowboy boots are located outside of (89) a mall in San Antonio, Texas. The boots are 40 feet tall. Suppose a painter at the top of a boot drops a paintbrush. The height of the brush, in feet, after x seconds, is modeled by y = -16x2 + 40. Write and solve a quadratic inequality that can be solved to find the time span for which the brush is above a second painter located 20 feet beneath the first painter. Round to the nearest hundredth of a second.

*14. Depreciation A truck is purchased for $18,500 on January 2, 2007. Its value (93) decreases 16% each year. In what year will its value first be less than $2,000?

656 Saxon Algebra 2 SSM_A2_NLB_SBK_L093.inddM_A2_NLB_SBK_L093.indd PagePage 657657 6/4/086/4/08 12:28:3412:28:34 PMPM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

_θr2 15. Multi-Step The formula for the area of a sector of a circle is A = , where θ is (88) 2 the central angle measure in radians and r is the radius. θ a. A landscape architect wants to create a flower bed in the shape of a sector of a r circular ring, as shown by the shaded region. Write a formula for the area S of a sector of a circular ring.

b. Solve the formula you wrote in part a for θ. R c. The landscape architect wants the area S of the flower bed to be 200 θ square feet, with R = 25 feet and r = 10 feet. Determine the necessary angle r measure θ and then convert it to the nearest degree.

_ln 8 16. Error Analysis A student rewrote the expression log8 (3x) as . (87) ln 3 + ln x a. Describe the student’s error. b. Rewrite the original expression correctly.

17. Analyze Consider the polynomial function f (x) = ax2 + bx + cx. Suppose that the (85) _ c polynomial is divisible by ( x - a ). Use synthetic division to show that a + b + c = 0. 18. Which of the following has a period of 2π ? (82) _x A y = sin (2πx) B y = sin (πx) C y = sin (x) D y = sin ( 2 ) *19. Estimate Thallium has a half-life of 73.1 hours. Estimate how long it will take for (93) 1 gram of thallium to decay to 0.26 gram. Explain your method.

Solve to the nearest tenth using the given restrictions. 20. sin θ = 0.95, for -90° ≤ θ ≤ 90° 21. cos θ = -0.181, for 180° ≤ θ ≤ 360° (67) (67)

Determine the domain and range of each function. 0, if x < 0 x, if 0 < x < 10 22. f (x) = 23. f (x) = (79) 3, if x ≥ 0 (79) 10, if x ≥ 10

Solve the following equations. 24. 3x4 = 12x3 - 9x2 + 6x 25. x3 = 2x4 - 5x2 + 4x (76) (76)

26. Generalize Given that a, b, c, and d are real numbers, explain why (x - a)(x - b) = (83) 0 and (cx - ca)(dx - db) = 0 have the same roots. What are the roots?

27. Given N(t) = 0.6, No = 1, k = 0.6372, write the decay function and solve for t. (81) 28. Write a quadratic equation whose roots are -5 and 4. (83) 29. Error Analysis A student says that the roots of the equation x4 - 81 = 0 are -3 (78) and 3. What complex roots were left out?

30. Probability The area under a normal curve is the probability that a randomly (80) selected data value from that data set falls within that interval. Suppose the weights of cereal boxes have a mean of 15.5 ounces with a standard deviation of 0.4 ounces. What is the probability that a randomly selected box of cereal weighs less than 14.7 ounces?

Lesson 93 657 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 658658 6/10/086/10/08 10:56:1910:56:19 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

LESSON Solving Rational Inequalities 94

Warm Up 1. Vocabulary To clear the fractions in a rational equation, multiply both (84) sides of the equation by the . _2x 2. Solve = 14 by using a graphing calculator. (84) x + 5 3. Solve x2 + 10x < -16 algebraically. (89)

New Concepts A rational inequality is an inequality that contains at least one rational expression. They can be solved algebraically or graphically. When a rational inequality is solved by using the LCD, the LCD could be positive or negative.

Example 1 Solving Rational Inequalities Using the LCD

_6 Math Reasoning Solve x + 2 < 8 by using the LCD. Analyze What is the SOLUTION Case 1: The LCD is positive. restricted value of the inequality in Step 1: The LCD is x + 2. Since it is positive, x + 2 > 0, so x > -2. Example 1? Step 2: Multiply both sides of the inequality by x + 2. 6 (x + 2) _ < 8 (x + 2) x + 2 6 < 8x + 16 Simplify. -10 < 8x Solve for x. x > -1.25 The solution must satisfy both x > -2 from Step 1 and x > -1.25 from Step 2, therefore, x > -1.25. Case 2: The LCD is negative.

Math Reasoning Step 1: The LCD is x + 2. Since it is negative, x + 2 < 0, so x < -2.

Justify Why did the Step 2: Multiply both sides of the inequality by x + 2. direction of the sign change in Case 2? (x + 2) _6 > 8(x + 2) Reverse the inequality sign. x + 2 6 > 8x + 16 Simplify. -10 > 8x Solve for x. x < -1.25 The solution must satisfy both x < -2 from Step 1 and x < -1.25 from Step 2, therefore, x < -2. Online Connection www.SaxonMathResources.com Combine the solutions from both cases: x < -2 or x > -1.25.

658 Saxon Algebra 2 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 659659 6/10/086/10/08 10:56:2710:56:27 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

As with solving quadratic inequalities, rational inequalities can be solved by finding critical values, or boundary points, and testing a point in each interval they form. Recall that one side must be equal to 0 to find the critical values.

Example 2 Solving Rational Inequalities by Finding the Values of Test Points

_x + 4

Solve x2 + x - 6 ≤ 0 by finding the values of test points. SOLUTION The right side is already 0. Find the boundary points. Factor. Find the values that make either the numerator or denominator 0. x + 4 __ ≤ 0 (x - 2)(x + 3) The numerator is 0 when x = -4. The denominator is zero when x = 2 or x = -3. The boundary points are -4, -3, and 2. This gives four intervals to test. Use < or > for the asymptotes of 2 and -3. These values are restricted because they make the fraction undefined. Math Reasoning Interval Test Value Is the test value a solution? Analyze Why is ≤ -5 + 4 instead of < used in __  0 x ≤ 4? (-5)2 + (-5) - 6 x ≤ -4 -5 - _1 ≤ 0 Yes 14 -3.5 + 4 __  0 (-3.5)2 + (-3.5) - 6 -4 ≤ x < -3 -3.5 _2 ≤ 0 No 11 0 + 4 _  0 02 + 0 - 6 -3 < x < 2 0 - _2 ≤ 0 Yes 3 5 + 4 __  0 (5)2 + (5) - 6 x > 2 5 _9 ≤ 0 No 24

The value of x is a solution when x ≤ -4 or -3 < x < 2.

Finding the actual value of a test point can become cumbersome. Notice that all that is really required from the test point is its sign. The intervals that contain solutions to the inequality have a negative sign because negative values are less than 0. The sign of the rational expression can be determined by finding the sign of the numerator and denominator, or by finding the sign of each factor.

Lesson 94 659 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 660660 6/10/086/10/08 10:56:3410:56:34 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

Example 3 Solving Rational Inequalities Using Sign Tables

_x - 6 a. Solve x + 2 > 0 by finding the sign of the numerator and denominator of the rational expression. SOLUTION The right side is already 0. Find the boundary points. The numerator is 0 when x = 6. The denominator is zero when x = -2. The boundary points are -2 and 6. This gives three intervals to test. They are listed in the top row with a test value below them. For each interval, find the signs of the numerator and denominator. Find the sign of the value of the rational expression for that interval by analyzing the signs of the numerator and denominator.

Hint x < -2 -2 < x < 6 x > 6

The quotient is negative Test value for x -3 09 when the signs are Numerator: x - 6 --+ different and positive when the signs are the Denominator: x + 2 -++ same. Value of rational expression +-+ The value of x is a solution when x < -2 or x > 6.

__(x - 5)

b. Solve (x + 1)(x - 4)(x - 7) ≥ 0 by finding the sign of each factor of the rational expression. SOLUTION The right side is already 0. Find the boundary points. The numerator is 0 when x = 5. The denominator is zero when x = -1, x = 4, or x = 7. The boundary points are -1, 4, 5, and 7. This gives five intervals to test. Use < or > for the undefined values (asymptotes) of -1, 4, and 7. Find the product of the “signs” of the factors to find the sign of the expression.

x < -1 -1 < x < 44 < x ≤ 55 ≤ x < 7 x > 7 Test value for x -2 0 4.5 6 10 x - 5 -- -++ x + 1 -+ +++ x - 4 -- +++ x - 7 -- --+ Value of rational +- +-+ expression

The value of x is a solution when x < -1 or 4 < x ≤ 5 or x > 7.

660 Saxon Algebra 2 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 661661 6/10/086/10/08 8:45:118:45:11 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

Example 4 Solving Rational Inequalities by Using a Graphing Calculator _x Solve x - 5 ≤ 4 by using a graphing calculator. _x SOLUTION Graph Y1 = x - 5 and Y2 = 4. Look for where the first graph intersects or is below the line. The entire left part of the rational expression is below the line. This occurs for all x-values to the left of the asymptote of x = 5, so one part of the solution is x < 5. Use the intersect command to find where the Graphing right part of the graph is below the line. The Calculator Tip _ intersection is at x = 6. 6 , so the second part of Press 2nd Window _ (Tblset) to change the the solution is x ≥ 6.6 . increments at which the x-values increase in the The solution_ of the inequality is x < 5 table. or x ≥ 6.6 . Check the solutions by viewing the table.

Example 5 Application: Cost per Person A group of students are sharing the cost of a $50 gift for their teacher. Write and solve an inequality to show the numbers of students whose participation would bring the cost per student to less than $4.25. SOLUTION _50 1. Understand The cost per student is x where x is the number of students. _50 2. Plan Solve x < 4.25. _50 3. Solve Graph Y1 = x and Y2 = 4.25. Find the point of intersection. The first graph is below the line for all x-values greater than the x- value of the point of intersection. Since x represents numbers of people, round up to 12 people. Since the number of people must be positive, disregard the negative solutions. Twelve or more students must participate to have a per per person cost of less than $4.25. 4. Check Look at the values in the table.

Lesson 94 661 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 662662 6/10/086/10/08 10:56:4910:56:49 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

Lesson Practice _7 a. Solve ≤ 3 by using the LCD. (Ex 1) x + 4 _x - 8 b. Solve 2 ≥ 0 by finding the values of test points. (Ex 2) x + 7x + 10 _x + 1 c. Solve < 0 by finding the sign of the numerator and denominator (Ex 3) x - 4 of the rational expression. __(x + 8) d. Solve > 0 by finding the sign of each factor of the (Ex 3) (x - 2)(x - 5)(x + 1) rational expression. _x e. Solve ≥ -2 by using a graphing calculator. (Ex 4) x + 6 f. A group of students are sharing the cost of a $75 gift for their teacher. (Ex 5) Write and solve an inequality to show the number of students whose participation would bring the cost per student to less than or equal to $6.25.

Practice Distributed and Integrated

Solve using the quadratic formula. Write the solution as complex numbers in standard form.

1. f (x) = x2 + 2x + 4 2. f (x) = x2 - x + 12 (65) (65)

3. Given P = 2l + 2w, find l. (88) _x2 - 3x - 18 *4. Multi-Step a. Factor both the numerator and denominator of 2 ≤ 0. (94) x - 16 b. Determine the intervals created by the critical values. c. The intervals for which test values are negative contain the solutions. What is the solution?

5. The average height of giant sunflowers is 6 feet tall and the standard deviation is (80) 3 inches. Find the z-score for a giant sunflower that is 5 feet 6 inches tall.

6. Multiple Choice What are the roots of the equation (4x2 - 4x + 1) - (78) (16x2 - 24x + 9) = 0? 2 2 1 1 3 A _ and - 2 B _ and 1 C _ and - _3 D _ and _ 3 3 2 4 2 4 *7. Government Suppose the citizens of counties in the state of Texas had the option to (94) donate an equal amount of money toward a one million dollar donation to be given to a charity. Write and solve an inequality to show the number of counties whose citizen participation would bring the cost per county to less than $6500. There are 254 counties in Texas.

8. Error Analysis A student said if a number cube is rolled and the results are recorded (80) in a histogram, a smooth curve drawn through the tops of the histogram would resemble a bell curve. Explain the error in the student’s thinking.

662 Saxon Algebra 2 SM_A2_NLB_SBK_L094.indd Page 663 7/12/08 2:45:23 AM user-s191 /Volumes/110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

_(x + 2) *9. Justify Explain why there are four intervals when solving by using test (94)(94) (x - 3)(x + 1) points.

10. Geometry The perimeter of an isosceles triangle is 200 inches. The base has a length (77)(77) of 92 inches. Find the measure of the vertex angle.

*11. Analyze If $500 is invested at 4% compounded monthly, then at the end of t years (93)(93) the investment will have the value A, where A is given by the expression _0.04 12t 500( 1 + 12 ) . What expression gives the value A if the compounding is daily instead of monthly?

12. What is the side length c in ABC if A = 32°, C = 67°, and b = 31 ft? (71)(71) *13. Error Analysis While attempting to solve an inequality, a student performed the step (93)(93) shown below. n log (0.5) < log 2 - log 20 log 2 - log 20 n < __ log 0.5 What is the error? What is the correct step?

14. An object is thrown upward from a height of 15 feet at an initial velocity of 35 feet (74)(74) per second. How long will it take the object to hit the ground?

15. Business The profit earned from selling tickets to a certain event is modeled by (89)(89) y = -1.25x2 + 66x - 720. If the cost per ticket will be a whole number amount, what range of ticket prices will earn a profit of at least $100?

*16. A school’s enrollment increases 6.5% a year and is currently 1100. How long will it (93)(93) take to reach 1500 students?

17. Probability A student has a spinner with four equal sections labeled 2, 4, 6, and 8. (68)(68) If he spins the spinner twice, find the probability that the first spin lands on 6 and the sum of the results is less than or equal to 10.

*18. Multiple Choice Which of the following has a period of 2π ? (90)(90) x A y = tan (2x) B y = tan (2πx) C y = 2 tan (x) D y = tan _ ( 2 ) *19. Graphing Calculator With the Table set to increase by 1, enter 15 + (x - 1)(-4) for (92)(92) Y1 to represent an arithmetic sequence whose first term is 15 and whose common difference is -4. Use the table to find the 6th and 15th terms.

20. Formulate Write a quadratic equation with roots of -1 and 5 so that the parabola (83)(83) of the related function opens up. Then write one so that the parabola opens down.

*21. Analyze When finding the nth term of an arithmetic sequence, when will the value (92)(92) added to the first term be positive and when will it be negative?

22. Given k(x) = 3 sin(x), identify the amplitude and period. (82)(82)

Lesson 94 663 SSM_A2_NLB_SBK_L094.inddM_A2_NLB_SBK_L094.indd PagePage 664664 6/10/086/10/08 10:57:0210:57:02 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L094

*23. Drafting A computer-aided drafting technician is drawing a bicycle wheel such (91) that the center is located at (6, 2) on the coordinate grid. Each unit on the grid represents one inch. What is the equation of the wheel given that it has a diameter of 28 inches?

24. Data Analysis Twelve names already numbered are listed below. Three must be (73) randomly chosen for a sample. What three names will be picked if numbers are chosen from the string of random numbers below? 15 Ann 16 Luke 17 Cassie 18 Finn 19 Jared 20 Tanela 21 Jude 22 Gunner 23 Drew 24 Halle 25 Colby 26 Anton 60201 60482 89824 08705 17913 46815 76221 25650 89337 53603 73597 04633

= _1 25. Given f 1 1 , find d2. (88) _ + _ d1 d2

26. Physics The equation for finding the period of a pendulum is a radical function (75) _L given by T = 2π √ 9.8 . The period is the amount of time it takes for the pendulum to complete a back-and-forth sweep. How much shorter does a pendulum need to be to have half the period?

27. Chemistry Uranium-238 is a naturally abundant isotope because of its long half- (81) life. Its half-life is about 4.5 billion years. About how long would it take for just -kt 1% of an initial amount to decay? (Use the natural decay function N(t) = N0e .)

28. Using the change of base formula, solve for y when x = 2 and x = 4 given that (87) _log x

y = log 2 .

Find the zeros of the polynomial function. 29. f (x) = 4x(7x - 3)(6x + 1) 30. f (x) = 16x3 - 24x2 + 6x (76) (76)

664 Saxon Algebra 2 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PagePage 665665 6/10/086/10/08 11:01:0611:01:06 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095

LESSON Factoring Higher-Order Polynomials 95

Warm Up 1. Vocabulary If f (x) = ax2 + bx + c is divisible by (x - d), then (x - d) is (35) a of f (x). 2. Use synthetic division to test if 6x3 - 5x2 - 34x + 40 is divisible by (x - 2). (51) 3. Use synthetic division to test if 20x3 - 53x2 - 122x + 56 is divisible by (51) (x - 2).

New Concepts A constant a is a root of polynomial P(x) if P(a) = 0. You can use the Reminder Theorem and the Factor Theorem to test for roots of a polynomial. Remainder Theorem The Remainder Theorem states: If the polynomial function P(x) is divided by x - a, then the remainder r is P(a).

Math Reasoning Factor Theorem Generalize Suppose a The Factor Theorem states: polynomial P(x) has no real roots. What does For any polynomial P(x),(x - a) is a factor of P(x) if and only if P(a) = 0. the Factor Theorem state about such a Use the Factor Theorem to test if a value is a root of a polynomial. polynomial? Example 1 Using the Factor Theorem to Test for Roots

a. Determine if x = 5 is a root of P(x). P(x) = 2x6 - 43x5 + 75x4 + 1765x3 - 857x2 - 22,542x - 30,240 SOLUTION Use synthetic division to test if r(5) = 0. Divide P(x) by (x - 5).

52-43 75 1765 -857 -22,542 - 30,240 10 -165 -450 6,575 28,590 30,240 2 -33 -90 1,315 5,718 6,048 0 Online Connection www.SaxonMathResources.com Since P(5) = r(5) = 0, 5 is a root of the polynomial.

Lesson 95 665 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PagePage 666666 6/10/086/10/08 11:01:1211:01:12 AMAM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095

b. Determine if x = -7 is a root of P(x). P(x) = 9x8 - 101x7 + 195x6 + 622x5 - 621x4 - 1565x3 - 1947x2 + 2772x + 2940 SOLUTION Use synthetic division to test if r(-7) = 0. Divide P(x) by (x + 7).

-7 9 -101 195 622 -621 -1565 -1947 2772 2940 -63 1148 -9401 61,453 -425,824 2,991,723 -20,928,432 146,479,620 9 -164 1343 -8779 60,832 -427,389 2,989,776 -20,925,660 146,482,560 Since P(-7) = r(-7) ≠ 0, -7 is not a root of the polynomial.

Example 2 Finding Roots of P(x) Find all the rational roots of P(x) = x5 + 4x4 - 10x2 - x + 6. SOLUTION By the Rational Root Theorem, the possible rational roots are ±1, ±2, ±3, ±6. Use synthetic division to test the roots. 11 4 0-10 -1 6 Since P(1) = 0, 1 is a root of the 15 5-5 -6 polynomial. 155-5 -6 0 P(x) = (x - 1)(x4 + 5x3 + 5x2 - 5x - 6). By the Rational Root Theorem, the possible rational roots of the quartic quotient are ±1, ±2, ±3, ±6. Use synthetic division to test the roots. 11 5 5-5 -6 Since P(1) = 0, 1 is a root of the polynomial 1 6 11 6 with multiplicity 2. 1 61160 P(x) = (x - 1)(x - 1)(x3 + 6x2 + 11x + 6). By the Rational Root Theorem, the possible rational roots of the cubic quotient are ±1, ±2, ±3, ±6. Use synthetic division to test the roots. 116116 -1 1 6 11 6 1718 -1 -5 -6 1 7 18 24 1560 Since P(-1) = 0, -1 is a root of the polynomial. P(x) = (x - 1)(x - 1)(x + 1)(x2 + 5x + 6) The last quotient is a quadratic that can be factored. P(x) = (x - 1)(x - 1)(x + 1)(x + 2)(x + 3) The roots are 1, -1, -2, and -3. The root 1 has a multiplicity 2.

666 Saxon Algebra 2 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PagePage 667667 6/10/086/10/08 8:59:558:59:55 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095

Math Reasoning Example 3 Application: Braking Distance Analyze Suppose that a When a car needs to be brought to an immediate stop, it takes time for polynomial P(x) has two real roots a and b. Could someone to step on the brake and time for the car to come to a complete P(x) be of degree 3? stop. During these times, the car is still moving. The stopping distance is how far the car travels during this time frame. A reasonable mathematical model for the stopping distance d, in feet, based on the car’s speed s, in miles per hour, is shown below. Use the Remainder Theorem to evaluate d(20), d(30), and d(55). d(s) = 0.05s2 + 2.2s SOLUTION Use synthetic division.

20 0.05 2.2 0 164 0.05 3.2 64 30 0.05 2.2 0 1.5 111 0.05 3.7 111 55 0.05 2.2 0 2.75 272.25 0.05 4.95 272.25 The stopping distances are d(20) = 64 ft, d(30) = 111 ft d(55) = 272.25 ft.

Lesson Practice Determine if x = 7 is a root of P(x). (Ex 1) a. P(x) = 2x6 - 43x5 + 75x4 + 1765x3 - 857x2 - 22,542x - 30,240 b. P(x) = 9x8 - 101x7 + 195x6 + 622x5 - 621x4 - 1565x3 - 1947x2 + 2772x + 2940 c. Find all the rational roots of P(x) = x2 + 4x2 + 1x - 6. (Ex 2) d. The height of an object thrown at 40 feet per second can be modeled by (Ex 3) h(t) = -16t2 + 40t where h is the height in feet and t is time in seconds. Use the remainder theorem to find the height of the object after 2 seconds.

Lesson 95 667 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PPageage 666868 66/3/08/3/08 7:42:107:42:10 PMPM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095

Practice Distributed and Integrated

Solve. 6 x2 + x - 6 7x 1. x - _ = 1 2. _ = 0 3. _ = 2 (84) x (84) x + 1 (84) 3x + 2

4. Does the graph of y = bx + k , through the points (3, -3) and (5, 0), model (57) exponential growth or decay? _1 *5. Solve 3 ≥ 5. Round to the nearest thousandth. (94) x 6. In a certain recipe, the amount of sugar is directly proportional to the amount of (8) flour. If 3 cups of sugar are used with 8 cups of flour, how many cups of sugar are used with 12 cups of flour?

*7. Error Analysis Two students were evaluating P(-4) for the polynomial P(x), but they (95) got different results. Which student made the mistake? P(x) = x6 + 4x5 - x4 - 6x3 - 8x2 + 2x + 8

Student A Student B -4 1 4 -1 -6 -8 2 8 4 1 4 -1 -6 -8 2 8 -4 0 4 8 0 -8 4 32 124 472 1856 7432 1 0 -1 -2 0 2 0 1 8 31 118 464 1858 7440 P(-4) = 0 P(-4) = 7440

_(x - 4)(x + 6) *8. Analyze When solving ≥ 0, which intervals will use strict inequalities (94) (x + 2)(x - 3) (< or >) and why?

*9. Multi-Step As of the 2000 U.S. census, Nevada had the fastest growing (64) population of all the states, and Connecticut had nearly the slowest (47th out of 50). The table below is based on U.S. census statistics.

Nevada Connecticut Population in 2000 (to nearest thousand) 1,998,000 3,406,000 Average annual increase (1990 –2000) 5.21% 0.36%

t a. Write a function (y1 = ab ), where y1 represents Nevada’s population t years after 2000. t b. Write a function (y2 = ab ), where y2 represents Connecticut’s population t years after 2000. c. Write and solve an equation to predict the year in which Nevada’s population will overtake Connecticut’s population.

Simplify each of the following. 10. 2 ln ex 11. x · ln e3 (81) (81)

668 Saxon Algebra 2 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PPageage 666969 66/3/08/3/08 7:42:167:42:16 PMPM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L095

12. Estimate A researcher visits a wildlife preserve and marks endangered African (73) elephants. He marks 6 elephants and releases them. On a return visit several months later, he comes across 16 elephants, and 4 of them are marked from the previous visit. Estimate the elephant population in the preserve.

13. Surveying A surveyor finds that the lengths between three stakes on the ground (77) are 125 feet, 182 feet, and 211 feet. He connects the stakes with string, forming a triangle. To the nearest tenth, what is the measure of the largest angle in this triangle? ⎧ -x , if x < 0 14. Determine the domain and range of the function f(x) = 4, if 0 ≤ x < 10 . (79) ⎨ ⎩18, if x ≥ 10

15. Research In a study of 413 men, the mean height was 174.1 centimeters and the (80) standard deviation was 7 centimeters. Assuming that the distribution is normal, approximate the z-score of a man whose height is 182 centimeters.

16. A set of values has a mean of 13 and a standard deviation of 0.625. Find the (80) z-score of a value of 10.25.

17. Justify Formulate a conjecture (statement) about the degree of a polynomial that (11) is a sum or difference of polynomials compared to the degree of the polynomials that are added or subtracted to get that sum or difference. Give examples to justify

your conjecture. (Hint: Name the polynomials P1, P2, P3, etc. to make it easy to refer to them.)

18. Geometry The area of a rectangular field is 100 meters. The field is 15 meters (23) longer than it is wide. Find the length and width of the field.

*19. Multiple Choice Which of the following polynomials has P(5) = 0? (95) A P(x) = 48x6 + 212x5 - 1098x4 - 5298x3 + 1174x2 + 22,350x + 18,900 B P(x) = 48x6 + 644x5 + 2754x4 + 2154x3 - 12,974x2 - 30,750x - 18,900 C P(x) = 48x6 + 868x5 + 6282x4 + 23,238x3 + 46,274x2 + 46,950x + 18,900 D P(x) = 48x6 - 268x5 - 818x4 + 4282x3 + 6254x2 - 14,790x - 18,900

20. Multiple Choice Identify which property or properties of real numbers are being (1) demonstrated. (2 · 9) · 5 = (2 · 5) · 9 A Commutative Property of Multiplication B Associative Property of Multiplication C Distributive Property D Both A and B

*21. Write a possible polynomial P(x) that fits the parameters P(0) = 0, P(2) = 20, (95) and P(-4) = 0.

3 _2x *22. Use the properties of logarithms to expand the expression ln (8x) + ln e . (87) ( 3 )

Lesson 95 669 SSM_A2_NLB_SBK_L095.inddM_A2_NLB_SBK_L095.indd PagePage 670670 6/4/086/4/08 1:27:081:27:08 PMPM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

23. Predict The table below shows the sales in dollars, y, made by a gift (13) shop x years after it opened.

Years 258 Sales $30,000 $75,000 $120,000 a. Plot the data on a coordinate grid and draw a line that fits the data.

b. Use the line to predict the total sales 15 years after the shop opened.

24. Error Analysis A student performs the following steps while finding the roots of (76) y = (x2 - 7)(3x2 + 4) - (x2 - 7)(8x - 3). What is his mistake? (x2 - 7)(3x2 + 4) - (x2 - 7)(8x - 3) = (x2 - 7)(3x2 - 8x + 1)

*25. Travel A driver on the Pennsylvania turnpike turned on the cruise control at (92) milepost 30, Warrendale, and kept it on until milepost 242, Harrisburg West, exactly 4 hours later. Show how to use an arithmetic sequence to find the speed at which the car traveled during those 4 hours.

26. Solve ABC. AB7 (71) 56°

14

C

*27. Write the equation for the circle centered at (3, 8) with a radius of 4. (91) 28. John saves $125 each month. Use the Distributive Property to mentally calculate (1) the amount of money that he will save in 18 months.

*29. Graphing Calculator Use a graphing calculator to graph the function (90) y = 3 tan(2x - 2π) - 4. Identify its period, undefined values, and phase shift.

*30. Write a quadratic equation whose root is -3. (83)

670 Saxon Algebra 2 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PPageage 667171 66/10/08/10/08 11:13:5011:13:50 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096

LESSON Using Polar Coordinates 96

_opposite side Warm Up 1. Vocabulary The trigonometric ratio is called the ______(46) hypotenuse ratio. 2. cos π = ______(46) 3. sin π = ______(46)

New Concepts A polar coordinate system in a plane is formed by a fixed P(r, θ) point O, called the pole (or origin), and a ray, called the r polar axis, whose endpoint is O. Each point in the θ Polar axis plane can be assigned polar coordinates as follows. O (Pole)

Polar Coordinates Every point P in the polar coordinate system has an ordered pair of polar coordinates (r, θ), where • r is the directed distance from O to P, and • θ is the −−directed angle measure counterclockwise from the polar axis to OP. _7π The point P(2, ) lies 2 units from the pole on the terminal _π 6 2 _7π side of the angles 6 . In a rectangular coordinate system, every point has exactly one ordered pair (x,y). But in a θ = _π polar coordinate system, every point has an infinite Q 6 O number of ordered pairs. The point P shown at π 123 0 _7π the right has coordinates 2, . But the angle 2, _7π ( 6 ) ( 6 ) P _7π _19π 7π with measure + 2π = also has terminal θ = _ 6 6 6 _19π ray OP , so point P also has coordinates (2, 6 ) . _3π And because r is a directed distance, point P also has 2 _π coordinates (-2, 6 ) . To understand this, notice that Some of the coordinates of P: _π _7π _19π _π Math Language 2, , 2, , and 2, point Q has coordinates (2, 6 ) ; the point with ( 6 ) ( 6 ) ( 6 ) _π A rectangular coordinates ( -2, 6 ) is point P, which is 2 units coordinate system is from the pole, on the ray opposite OQ . also called a Cartesian coordinate system. In general, the point (r, θ) can be represented as (r, θ ± 2nπ) or (-r, θ ± (2n + 1)π), where n is any integer. To relate the Cartesian and Polar coordinate y-axis systems, let the pole system coincide with the P(x, y) origin of a Cartesian (rectangular) coordinate P(r, θ) r system and let the polar axis coincide with the y Online Connection positive x-axis. Then θ x- or Polar _x _y _y x axis www.SaxonMathResources.com cos θ = r , sin θ = r , and tan θ = x . Origin or Pole

Lesson 96 671 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PPageage 667272 66/11/08/11/08 5:07:465:07:46 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096

To convert coordinates, use the following equations.

Converting Coordinates Polar to Cartesian Cartesian to Polar x = r cos θ y tan θ = _ y = r sin θ x r2 = x2 + y2

Example 1 Converting Polar Coordinates to Cartesian Coordinates

_π y a. Convert (2, 2 ) to Cartesian coordinates. 2 π (r, θ) = 2, _ SOLUTION ( 2 ) π 1 x = r cos θ = 2 cos _ = 2(0) = 0 (x, y) = (0, 2) 2 O x -1 1 2 _π y = r sin θ = 2 sin = 2(1) = 2 -1 2 The Cartesian coordinates are (0, 2).

π Convert 2, - _ to Cartesian coordinates. b. ( 6 ) SOLUTION y π 1 - _ is equivalent to -30º. 6 O x

√ 3 -1 1 2 x = r cos θ = 2 cos - _π = 2 _ = √3 ( ) ( 2 ) -1 6 π_ (r, θ) = (2, - ) -2 6 y = r sin θ = 2 sin - _π = 2 - _1 =-1 ( 6 ) ( 2 ) (x, y) = (√3, 1) The Cartesian coordinates are ( √3 , -1) .

Example 2 Converting Cartesian Coordinates to Polar Coordinates

a. Convert (-3, -3) to polar coordinates. SOLUTION y Hint O x _y _-3 1 π -3 -2 -1 - _ tan θ = x = = 1 Tan 1 = 4 , and θ -3 terminates in quadrant III, (x, y) = (-3, -3) -1 so one value of θ is Since (-3, -3) lies in quadrant III, θ terminates _π _5π _5π + π = . in quadrant III. Therefore, one value of θ is . 5π -2 4 4 4 (r, θ) = 3 √2, _ ( 4 ) r2 = x2 + y2 =(-3)2 +(-3)2 = 18 -3 r = √18 = 3 √2 √ _5π One ordered pair of polar coordinates is ( 3 2 , 4 ) .

672 Saxon Algebra 2 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PPageage 667373 66/10/08/10/08 8:21:008:21:00 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096

b. Convert (0, -1) to polar coordinates. y SOLUTION O x y -1 tan θ = _ = _ , which is undefined. Since θ terminates 1 2 3 x 0 _3π -1 (x, y) = (0, -1) on the negative y-axis, one value of θ is 2 . 3π 2 2 2 2 2 -2 (r, θ) = 1, _ r = x + y = (0) + (-1) = 1 ( 2 ) r = 1 -3 _3π One ordered pair of polar coordinates is ( 1, 2 ) . A polar equation is an equation for a curve with coordinates r and θ. There are two basic polar equations r = k and θ = α, where k is a positive constant and α is a constant. The graph of r = k forms a circle and the graph of θ = α forms a line through the origin. These graphs can be identified by converting these types of basic polar equations into the Cartesian coordinate system.

Example 3 Graphing Polar Equations

a. Graph r = 2. _π 2 SOLUTION A circle with all points 2 units from the pole.

123 Check Convert to a Cartesian equation. π 0 r = 2 r2 = 4 Square both sides. 2 2 2 2 2 x + y = 4 r = x + y _3π 2 This is the Cartesian equation of the same circle. π Graph θ = _ . b. 6 SOLUTION _π Math Reasoning 2

Analyze If a is a positive The graph consists of all points on the line that π constant, describe the _ makes an angle of 6 radian with the positive x-axis. graphs of polar equations of the following forms. Check Convert to a Cartesian equation. π 123 0 r = a and θ = a y tan θ = _ Definition of tan θ. x π y tan _ = _ Substitute θ = _π . 6 x 6 _3π 2 √3 y _ = _ Evaluate tan _π . 3 x 6 √3 y = _ x Solve for y. 3

Lesson 96 673 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PagePage 674674 6/4/086/4/08 1:42:301:42:30 PMPM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

To graph a polar equation that has two variables, make a table and plot points. Below is a table of values that could be used to begin the graph of the polar equation r = 2θ.

_π _π π θ 0 _ 6 3 2 _π 2π r(θ) = 2θ 0 ≈ 1.05 _ ≈ 2.09 π ≈ 3.14 3 3

Graphing polar equations by plotting points is often a long process. The equation in the table above is graphed by calculator in Example 4a.

Graphing Example 4 Graphing Polar Equations on a Graphing Calculator Calculator Tip Graph each polar equation on a graphing calculator. Make sure your calculator is in polar a. r = 2θ mode by pressing the mode key. SOLUTION Use the window indicated below. θ min = 0 X min = -36 θ max = 4π X max = 36 Y min = -24 θ step = _π 24 Y max = 24

b. r = 1 - sin θ Math Language SOLUTION The graphs in Example 4 have these names: Use the window indicated below. a. Archimedean spiral b. cardioid θ min = 0 X min = -3 c. rose θ max = 2π X max = 3 π Y min = -3 θ step = _ 24 Y max = 1

c. r = 5 cos 2θ SOLUTION Use the window indicated below. θ min = 0 X min = -9 θ max = 2π X max = 9 π Y min = -6 θ step = _ 24 Y max = 6

674 Saxon Algebra 2 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PagePage 675675 6/4/086/4/08 1:45:491:45:49 PMPM pripri //Volumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TEVolumes/ju114/MHGL153/Indd%0/Grade_12/TOC-TE

Example 5 Application: Cell Phone Tower A planned cell phone tower will service an area with a B 5-mile radius. It is to be located 3 miles east and 4 miles north of a highway intersection. Write and graph a polar equation 4 mi A to show the boundary of the region that will be serviced. Place the highway intersection at the pole. 3 mi SOLUTION The boundary is a circle with radius 5 and center (3, 4) on a Cartesian coordinate system, so the Cartesian equation is (x - 3)2 + (y - 4)2 = 52. (x - 3)2 + (y - 4)2 = 25 x2 - 6x + 9 + y2 - 8y + 16 = 25 Expand the binomials. x2 + y2 - 6x - 8y = 0 Simplify. r2 - 6r cos θ - 8r sin θ = 0 x2 + y2 = r2, x = r cos θ, and y = r sin θ. r(r - 6 cos θ - 8 sin θ) = 0 Factor. r - 6 cos θ - 8 sin θ = 0 r ≠ 0, so r - 6 cos θ - 8 sin θ = 0. r = 6 cos θ + 8 sin θ Add 6 cos θ + 8 sin θ to both sides. The polar equation is r = 6 cos θ + 8 sin θ. Graph the polar equation on a calculator. Use the window indicated below. θ min = 0 X min = -3 θ max = 2π X max = 15 π Y min = -2 θ step = _ 24 Y max = 10

Lesson Practice a. Convert (3, π) to Cartesian coordinates. (Ex 1) _2π b. Convert 1, to Cartesian coordinates. (Ex 1) ( 3 ) c. Convert ( 2 √3 , -2) to polar coordinates. (Ex 2) d. Convert (-4, 0) to polar coordinates. (Ex 2) e. Graph r = 3. (Ex 3) _3π f. Graph θ = . (Ex 3) 4 g. Graph r = θ on a graphing calculator. (Ex 4) h. Graph r = 1 + cos θ on a graphing calculator. (Ex 4) i. Graph r = 4 sin 3θ on a graphing calculator. (Ex 4) j. A planned cell phone tower will service an area with a 2-mile radius. It (Ex 5) is to be located 2 miles north of an office building. Write and graph a polar equation to show the boundary of the region that will be serviced. Place the location of the building at the pole.

Lesson 96 675 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PPageage 667676 66/10/08/10/08 11:23:4211:23:42 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096

Practice Distributed and Integrated

2 1. Multi-Step a. Use the change of base formula to convert log1000(10x) to base 10. (87) b. Evaluate when x = 100.

2. Geography The city of Madera, California, is 95 miles due southeast of Stockton, (52) which is 95 miles due southwest of South Lake Tahoe. How far would you have to fly from Madera to South Lake Tahoe, and in what direction? _π *3. Convert (-3, ) to Cartesian coordinates. (96) 2 4. A taxicab company that wants to know if customers are satisfied has each driver (73) survey three customers during the day. Is the sample biased? Explain.

*5. Justify Suppose that for polynomials P1(x) and P2(x), there are constants a and b (95) such that P1(a) = P2(b) = 0. Is it correct to conclude that b is a root of P1(x) and a

is a root of P2(x)? Explain.

Solve and graph the following compound inequalities. 6. 3t > 18 or t - 3 < 0 7. -(h - 2) > 7 or -8 ≥ -2h (10) (10)

8. Statistics The table shows the number of home runs (1) Year 1 Year 2 Year 3 hit by a softball player over a three-year period. Use Home runs 81517 properties of real numbers to mentally calculate the average annual home runs that she hit during this period.

Determine the domain and range. _1 _3 3 9. y = x 3 - 1 10. f(x) = √x + 2 + √ x - 1 (75) 4 (75)

*10. If $1,000 is invested at 7% compounded continuously, how long would it take for (93) the value of the investment to reach $3,000?

11. Juggling A juggler tosses a ball into the air. The ball leaves the juggler’s (74) hand 4 feet above the ground and has an initial velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

12. In LMN, LM = 14, MN = 8, and m∠M = 84°. Find LN. (77) 13. Simple Interest The simple interest formula is A = P + Prt, where P is the (88) principal, or original amount invested, r is the annual simple interest rate, and A is the value of the investment at the end of t years. Solve the formula for t. Then determine how many years it will take for an investment to double in value at a 5% annual simple interest rate.

14. Geometry Derive the function that can be used to find the area of an isosceles (82) triangle with base a and congruent sides measuring b. Each base angle measures θ. Express in simplified form.

676 Saxon Algebra 2 SSM_A2_NLB_SBK_L096.inddM_A2_NLB_SBK_L096.indd PPageage 667777 66/10/08/10/08 8:21:118:21:11 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L096

*15. Multiple Choice Which equation represents a circle centered at (–5, 4) with a radius (91) of 3? A (x - 5)2 + (y + 4)2 = 3 B (x - 5)2 + (y + 4)2 = 9 C (x + 5)2 + (y - 4)2 = 3 D (x + 5)2 + (y - 4)2 = 9

16. National Parks The Surprise Arch, in Arches National Park, has a span of about (83) 60 feet and a height of about 50 feet. Write a quadratic function to approximate the arch.

*17. Find the roots of y = x9 - x7 - 8x6 - 16x5 + 8x4 + 16x3 + 128x2 - 128. (95) 18. Formulate Solve _1 + _1 = _1 for c. (84) a b c 19. Find the roots of f(x) = 0.15x3 - 2.85x2 + 14.85x - 12.15 using the Rational (85) Root Theorem.

*20. Graphing Calculator Use a graphing calculator to graph y = cos(7x) and compare its (86) period to that of the parent function.

y ≤ -x2 + 4x + 8 21. Multiple Choice Which is not a solution of ? (89) y < -3x + 1 A (0, 0) B (0, -3) C (1, -6) D (2, -2) 22. Let f(x) = tan(5x + 5) and g(x) = 3x + 5. Find the period of f(g(x)). (90) *23. A kayaker paddles with the current for 7 miles, turns around, and paddles against (84) the current for 7 miles. The average paddling rate is 3 miles per hour. Write an inequality to represent the speed of the current if the total time is less than 7 hours.

24. Error Analysis To rewrite ln(ex)3, a student wrote (81) ln (ex)3 = ln(3 · (ex)) = ln 3 + ln (ex) = ln3 + ln e + ln x = ln 3 + ln x + 1. What is the error? Rewrite ln (ex)3 as a sum or difference of terms correctly.

*25. Write Explain why every point in a polar coordinate system has more than one (96) set of polar coordinates.

*26. Savings During week 1, a customer opened a bank account with $450. Each week (92) thereafter, the customer deposits $50. How much will the customer have in the account after the deposit is made during week 52?

27. Error Analysis A student says that the roots of the equation 4x3 - 36x = 0 are -3 (78) and 0. What is the error?

28. Verify Use FOIL to verify the difference of two squares formula a2 - b2 = (a + b) (23) (a - b).

29. Analyze How are z-scores of –0.5 and 0.5 alike? How are they different? (80) *30. Cell Phone Tower A planned cell phone tower will service an area with a 6 mile (96) radius. It is to be located 6 miles west of a small town. Write a polar equation of the boundary of the region that will be serviced. Place the location of the town at the pole. Graph the equation on a calculator.

Lesson 96 677 SSM_A2_NLB_SBK_L097.inddM_A2_NLB_SBK_L097.indd PPageage 667878 44/18/08/18/08 5:51:185:51:18 PMPM useruser //Volumes/ju109/HCAC061/SM_A2_SBK_setup%0/SM_A2_workingVolumes/ju109/HCAC061/SM_A2_SBK_setup%0/SM_A2_working ffile/Workingile/Working

LESSON Finding Geometric Sequences 97

Warm Up 1. Vocabulary Each number in a sequence is called a . (92) 2. True or False: The sequence {2, 4, 8, 16} is arithmetic. (92) 3. Simplify 5(-3)4. (2)

New Concepts The sequence {3, 6, 12, 24, 48} is not arithmetic because the differences between consecutive terms are not the same. However, the ratio of any two consecutive terms is the same.

_48 = 2, _24 = 2, _12 = 2, _6 = 2 24 12 6 3 This makes the sequence geometric. In a geometric sequence, the ratio of successive terms is a constant other than 1. This constant is called the common ratio. It is found by dividing a term by its previous term and can be used to extend an infinite sequence.

Example 1 Identifying and Using the Common Ratio Find the common ratio of each geometric sequence and use it to find the next three terms.

3 3 a. 6, 3, _ , _ , … 2 4 1 SOLUTION Divide a term by a previous term: 3 ÷ 6 = _ . 2 1 Find the next three terms by multiplying by _ : 2 3 1 3 3 1 3 3 1 3 _ · _ = _ , _ · _ = _ , and _ · _ = _ . 4 2 8 8 2 16 16 2 32 3 3 3 The next three terms are _ , _ , and _ . 8 16 32 b. -4, 8, -16, 32, … SOLUTION Divide a term by a previous term: 32 ÷ -16 = -2. Find the next three terms by multiplying by -2: 32 · -2 = -64, -64 · -2 = 128, and 128 · -2 = -256. The next three terms are -64, 128, and -256.

Hint An explicit formula can be determined to find the nth term of a geometric sequence. For a ≠ 0, a0 = 1. For the geometric sequence {2, 6, 18, 54, 162, …}, 0 1 2 3 4 a1 = 2(3) , a2 = 2(3) , a3 = 2(3) , a4 = 2(3) , a5 = 2(3) .

678 Saxon Algebra 2 Math Reasoning Geometric Sequences Formulate What is a The nth term of a geometric sequence is given by recursive formula for a n-1 geometric sequence? an = a1 r , where r is the common ratio.

Example 2 Finding the nth Term of a Geometric Sequence

a. Find the tenth term of the geometric sequence 3, 12, 48, 192, …

n-1 SOLUTION Find r: 12 ÷ 3 = 4. Then use the formula a n = a1 r . 10-1 n a a10 = 3(4) Substitute 10 for , 3 for 1 , and 4 for r.

a 10 = 3(262,144) Simplify.

a 10 = 786,432

b. Find the 8th term of the geometric sequence 810, -270, 90, -30, ... _1 n-1 SOLUTION Find r: -30 ÷ 90 = - . Then use the formula a n = a r . 3 1 8-1 _1 n a _1 r a8 = 810 - Substitute 8 for , 810 for 1 , and - for . ( 3) 3 _1 a8 = 810 - Simplify. ( 2187) _810 _10 a8 = - = - 2187 27

n-m The formula a n = am r can be used to find a term when given either a term and the common ratio or any two terms of the sequence.

Example 3 Finding the nth Term Given a Term and r The fifth term of a geometric sequence is 567. The common ratio is 3. Find the ninth term.

n-m SOLUTION In the formula a n = am r , replace both instances of m with 5, replace n with 9, and replace r with 3.

9-5 a9 = a5 r

4 a9 = 567(3)

a9 = 567(81)

a9 = 45,927 When r must first be found, there may be two cases, depending on whether an Online Connection even root or an odd root must be taken. This could lead to two possible values www.SaxonMathResources.com for the nth term.

Lesson 97 679 Example 4 Finding the nth Term Given Any Two Terms

a. Find a1 of a geometric sequence given that a4 = 32 and a9 = 1024.

n-m SOLUTION In the formula a n = am r , replace m with 4 and replace n with 9.

9-4 a9 = a4r 5 1024 = 32r Substitute a9 with 1024 and a4 with 32. 32 = r5 Solve for r. 2 = r Take the fifth root of each side.

Now find a1. Use either of the known terms. 4-1 n 32 = a1(2) Substitute 32 for an and 4 for . a 32 = a1(8) Solve for 1.

4 = a1 The first term of the sequence is 4. Check

8 Check by using a1 to find the ninth term: a9 = 4(2) = 4(256)= 1024.

b. Find a7 of a geometric sequence given that a3 =-36 and a5 =-324.

n-m SOLUTION In the formula a n = amr , replace m with 3 and replace n with 5.

5-3 a5 = a3r

2 - 324 =-36r Substitute a5 with -324 and a3 with -36. 9 = r2 Solve for r. ±3 = r Take the square root of each side.

Now find a7. Consider both r = 3 and r =-3. Case 1: r = 3 Case 2: r = -3

7 - 3 7 - 3 a7 = a3(3) a7 = a3(-3)

4 4 a7 = -36(3) a7 = -36(-3)

a7 = -36(81) a7 = -36(81)

Math Reasoning a7 = -2916 a7 = -2916

Verify Use the formula In this example, both possible values of r give the same value for a7. The to show that a4 has two seventh term of the sequence is -2916. possible values. Check

a3 a4 a5 a6 a7 r = 3 -36 -108 -324 -972 -2916 r = -3 -36 108 -324 972 -2916

680 Saxon Algebra 2 Example 5 Application: Salary An employee’s salary is structured so that he earns $43,400 in the first year with a 3.5% raise each year thereafter. How much can the employee expect to earn in his fifteenth year? SOLUTION The salaries each year form a geometric sequence where n-1 r = 1.035. Substitute 15 for n, 43,400 for a1, and 1.035 for r in an = a1 r .

Math Reasoning 15-1 a15 = 43,400(1.035) Justify Why is r equal to 1.035? a15 = 70,251.34 Round to the hundredths place. The employee’s salary in the fifteenth year will be $70,251.34.

Check Use a graphing calculator.

Lesson Practice Find the common ratio of each geometric sequence and use it to find the next three terms. a. 2, 0.2, 0.02, 0.002, … b. -200, -40, -8, - _8 … (Ex 1)1)(Ex (Ex 1) 5 c. Find the ninth term of the geometric sequence -1, 5, -25, 125, … (Ex 2)(Ex 2) _1 d. Find the eleventh term of the geometric sequence 16, 4, 1, , … (Ex 2)(Ex 2) 4 _1 1 e. The fourth term of a geometric sequence is 1 . The common ratio is - _ . (Ex 3)(Ex 3) 4 2 Find the eighth term.

f. Find a1 of a geometric sequence given that a3 = 128 and a6 = 8192. (Ex 4)4)(Ex

g. Find a9 of a geometric sequence given that a4 = 24 and a6 = 96. (Ex 4)4)(Ex h. An employee’s salary is structured so that he earns $35,890 in the (Ex 5)(Ex 5) first year with a 4.25% raise each year thereafter. How much can the employee expect to earn in his tenth year?

Practice Distributed and Integrated

*1. Given V = πr2h, find h. (88)(88) 2. Write A survey asks students, “Who would you vote for in the next school (73)(73) election, the relatively unknown sophomore Eileen Johnson, or the popular Zack Jennings?” Explain the bias in the question.

Simplify the following expressions.

log 8 x x 3. 8 4. log3 81 (72)(72) (72)

Lesson 97 681 SSM_A2_NLB_SBK_L097.inddM_A2_NLB_SBK_L097.indd PPageage 668282 44/18/08/18/08 5:51:255:51:25 PMPM useruser //Volumes/ju109/HCAC061/SM_A2_SBK_setup%0/SM_A2_workingVolumes/ju109/HCAC061/SM_A2_SBK_setup%0/SM_A2_working ffile/Workingile/Working

*5. Multi-Step The value of a car in its first year was $18,900. Each year thereafter, (97) the value was 85% of what it was the previous year. The values each year form a geometric sequence. a. Write a formula to find the value of the car in the nth year. b. Find the value of the car in year 6.

6. What number can be added to both sides of the equation x2 + 18x + 36 = 0 to (Inv 6) make it a perfect square?

7. Analyze How can you tell that 4 + √x = 1 will have no solutions? (70)

⎡-6 2⎤ ⎡11 2⎤ Find each of the following matrix operations for A = ⎢ and D = ⎢ . ⎣-1 -3⎦ ⎣ 8 3⎦ 8. Find A + D. 9. Find D - 3A. (5) (5)

*10. Find the fifteenth term of the geometric sequence 2, 6, 18, … (97) 11. Error Analysis Which student made an error in determining the number of solutions (74) of -3x2 + 6x = 4? What was the error? b2 - 4ac b2 - 4ac 62 - 4(-3)(4) 62 - 4(-3)(-4) Student A: 36 + 48 Student B: 36 - 48 84 -12 2 real solutions 2 complex solutions

*12. Estimate Write approximate polar coordinates for the point with Cartesian (96) coordinates (6, 6.2). Explain your method.

13. Geometry A student is told to draw a rectangle so that the perimeter is 48 units and (62) the area is 150 square units. The equation 24x - x2 = 150 gives the values of x that meet these criteria where x is the length of the rectangle. Solve the equation. Describe your findings.

*14. City Planning A park lawn is a square. A diagonal sidewalk is planned, shown as ___ B (96) AB in the diagram. Using point A as the___ pole and AC as the polar axis, write a polar equation of the line that contains AB . A C

15. Coordinate Geometry Graph {(2, 4), (5, 6), (–1, 0), (7, 10)}. Determine if this relation (22) is a function. Determine if the graph is continuous, discontinuous, and/or discrete. _1 16. Model Graph the square root function and its inverse. y = √x - 4 (75) 3 17. National Highways In Texas, the speed limit on state highways at night is 5 miles per (84) hour less than what it is during the day. A driver drove 350 miles during the day and 195 miles at night, with the cruise control set to the appropriate speed limit. The total driving time was 8 hours. Write and solve a rational equation to find the speed limit during the day.

682 Saxon Algebra 2 SSM_A2_NLB_SBK_L097.inddM_A2_NLB_SBK_L097.indd PPageage 668383 66/2/08/2/08 11:17:5711:17:57 PMPM elhielhi //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L097Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L097

*18. Multiple Choice The second term of a geometric sequence is 16 and the fourth term (97) is 1. What is the common ratio? 1 1 A _ B ± _1 C _ D ± _1 16 16 4 4 19. Analyze Consider the polynomial function f(x) = ax4 + bx3 + cx2 + dx + e. (85) Suppose that the polynomial is divisible by (x - 1). Use synthetic division to show that a + b + c + d + e = 0.

20. Physics Two wheels spin in the same direction. The larger wheel (86) completes a cycle in 3 seconds and the smaller wheel completes a cycle 6 4 in 5 seconds. The two squares attached to the wheels are facing each other at time t = 0. How many seconds elapse before they are back in sync? 21. Find the common difference of the arithmetic sequence -62, -67, -72, -77, … (92)

2 4 y + 12 > x 1 *22. Graphing Calculator Graph - _ x ≥ y on your graphing calculator. Name two (89) 2 y ≥ -2 points in the solution set.

23. Multiple Choice Which of the following has a period of 7π? (82) 2 x A y = cos(7x) B y = cos _ x C y = cos(3.5x) D y = cos _ ( 7 ) ( 7 ) 24. Given g(x) = 0.5sin(2x), identify the amplitude and period. (82) 25. A set of values has a mean of 15 and a standard deviation of 1.5. Find the percent (80) of values above 18.

⎧ 9 - 5x, if x < -0.1   2, if - 0.1 ≤ x ≤ 2 26. Evaluate the piecewise function f(x) = ⎨ for x = -0.4, x (79)  x3 - x, if 2 < x ≤ 6 = 0, and x = 6.  ⎩ 3 - x2, if x > 6

*27. Medicine Cobalt-60 is an isotope that is used in radiation therapy for cancer (93) patients. It has a half-life of about 5.3 years. How long will it take for 1 gram of -kt cobalt-60 to decay to 0.9 gram? (Use the natural decay function N(t) = N0 e . )

28. Boating A boat leaves a dock and travels in a straight path for 6 miles at an angle (77) 15° north of east. Another boat leaves the same dock and travels in a straight path for 11 miles at an angle 65° south of east. If the second boat puts down an anchor and the first boat travels directly toward the second, estimate the time it will take the first boat to reach the second boat if the first boat travels at a constant rate of 8 miles per hour.

*29. The endpoints of a diameter of a circle are located at (-10, -2) and (2, -2). (91) Write an equation for this circle.

4 30. Convert log6 (5x) to base e. Then evaluate when x = 3. (87)

Lesson 97 683 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PPageage 668484 66/11/08/11/08 1:12:101:12:10 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

LESSON Making Graphs and Using Equations 98 of Ellipses

Warm Up 1. Vocabulary The graph of x2 + y2 = r2 is a . (91) 2. What figure does the equation (x - 1)2 + (y - 1)2 = 52 form? (91) 3. Where does x2 + y2 = 72 touch the x-axis? (91) 4. If c2 = a2 - b2, which of these is the value of c if a = 13 and b = 5? (2) A c = ±12 B c = ± √194 C c = ±144 D c = ±194 5. If c2 = a2 - b2, find c if a = 4 and b = 3. (2)

New Concepts An ellipse is the set of all points P in a plane such that the sum of the

distance from P to two fixed points F1 and F2 is constant. The two fixed

points, F1 and F2, are called the foci. An ellipse has two axes. The major axis is the longer axis of the ellipse and passes through the foci. The endpoints of the major axis are the vertices of the ellipse. The minor axis is the shorter axis Math Reasoning of the ellipse. and its endpoints are the co-vertices. The major and minor axes Model Take a length of are perpendicular and their point of intersection is the center of the ellipse. string and thumbtack 2 2 2 2 both ends to a sheet of x y x y _ + _ = 1. _ + _ = 1. paper so that the string a2 b2 b2 a2 is not taut. Take a pencil y and use the string as a guide to drawing an y Minor axis ellipse. What role do the (0, b) (0, a) thumbtacks play? (–a, 0) c Minor axis (–c, 0) (c, 0) x (0, ) (-b, 0) (b, 0) x O (a, 0) Major axis -c Major axis (0, –b) (0, ) (0, -a)

Vertices: (±a, 0) Vertices: (0, ±a) Covertices: (0, ±b) Covertices: (±b, 0) Foci: (±c, 0) Foci: (0, ±c)

Standard Form of the Equation of an Ellipse: Center at (0, 0) Major Axis Horizontal Major Axis Vertical x2 y2 y2 x2 _ + _ = 1 _ + _ = 1 a2 b2 a2 b2 Vertices: (a, 0), (-a, 0) Vertices: (0, a), (0, -a) c c c c Online Connection Foci: ( , 0), (- , 0) Foci: (0 ), (0, - ) www.SaxonMathResources.com Co-vertices: (0, b), (0, -b) Co-vertices: (b, 0), (-b, 0)

684 Saxon Algebra 2 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PPageage 668585 66/11/08/11/08 5:33:265:33:26 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

Example 1 Writing an Equation of an Ellipse

y Write an equation in standard form for the ellipse 4 with center (0, 0). Major axis SOLUTION Hint O x -2 2 a always corresponds to Step 1: Choose the appropriate standard form. the major (longer) axis of x2 y2 the ellipse. _ + _ = 1 The horizontal axis is longer. Minor axis a2 b2 -4 Step 2: Identify the values of a and b. a = 4 The vertex (4,0) gives the value of a. b = 2 The co-vertex (0,2) gives the value of b. Step 3: Write the equation. x2 y2 _ + _ = 1 Substitute the values of a and b into the equation. 42 22 _x2 y2 The equation of the ellipse is + _ = 1. 42 22 There is an important relationship among a, b, and c: c2 = a2 - b2. This relationship can be used to find the foci of an ellipse. The eccentricity, e, of an ellipse is a measure of its curvature. Eccentricity is _c defined as a, where c is the distance from the center to a focus and a is the distance from the center to a vertex.

Example 2 Graphing an Ellipse Centered at the Origin Graph the following equation. Find the vertices, co-vertices and the foci. Calculate the eccentricity e. 25x2 + 9y2 = 225 SOLUTION

Step 1: Write the equation in standard form by (0, 5) y dividing both sides by the constant term. 4 (0, 4) x2 y2 y2 x2 _ + _ = 1 _ + _ = 1 2 9 25 2 2 5 3 (-3, 0) O (3, 0) x Step 2: Find the values of a, b, and c. -4 -2 2 4 -2 From the equation, a = 5 and b = 3. (0, -4) Use the equation c2 = a2 - b2 to find c. -4 (0, -5) c2 = 52 - 32 c = 4 The vertices are (0, ±5), the co-vertices are (±3, 0), and the foci are (0, ±4). c 4 The eccentricity is _ = _. a 5

Lesson 98 685 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PPageage 668686 66/11/08/11/08 5:33:395:33:39 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

Ellipses can be translated so that the center is not the origin. Standard Form of the Equation of an Ellipse: Center at (h, k) Major Axis Horizontal Major Axis Vertical (x - h)2 (y - k)2 (y - k)2 (x - h)2 _ + _ = 1 _ + _ = 1 a2 b2 a2 b2 Vertices: (h + a, k), (h - a, k) Vertices: (h, k + a), (h, k - a) Foci: (h + c, k), (h - c, k) Foci: (h, k + c), (h, k - c) Co-vertices: (h, k + b), (h, k - b) Co-vertices: (h + b, k), (h - b, k)

Example 3 Graphing Ellipses Not Centered at the Origin Graph the following equation. Find the center, vertices, co-vertices and the foci. Calculate the eccentricity e.

9(x - 2)2 + 64(y - 5)2 = 576

Math Reasoning SOLUTION Analyze Why is the Step 1: Write the equation in standard form by dividing both sides by the equation of an ellipse not a function? constant term. (x - 2)2 (y - 5)2 (x - 2)2 (y - 5)2 _ + _ = 1 _ + _ = 1 64 9 82 32

Step 2: Find the values y of a, b, c, h, and k. Major axis 12 The graph has a Minor axis (2, 8) a = 8 horizontal axis so, b = (2, 5) 3 (-6, 5) (10, 5) c = √82 32 = 7.416 from the equation, 4 a = 8, b = 3, h = 2, O x _32 (2, 2) e = 1 = 0.927 and k = 5. -4 4 8 √ 82 Use the equation c2 = a2 - b2 to find c. c2 = 82 - 32 c = √55 ≈ 7.416 c √55 _ _ The eccentricity is a = 8 ≈ 0.927. Step 3: Find the translated center, vertices, co-vertices, and foci. Center: (2, 5) Vertices: (2 + 8, 5),(2 - 8, 5) (10, 5), (-6, 5) Foci: (2 + √55 , 5), (2 - √55 , 5) Co-vertices: (2, 5 + 3), (2, 5 - 3) (2, 8), (2, - 2)

686 Saxon Algebra 2 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PagePage 687687 6/12/086/12/08 3:45:563:45:56 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

y The geometric definition of an ellipse is the locus of 4 points such that the sum of distances from any point r r 1 2 on the ellipse to the foci is constant. O x -8 -4 4 8 r1 + r2 = 2a -4

Example 4 Application: Planetary Orbits

All the planets have elliptical orbits, but each has Planet e a different eccentricity. The table lists each planet’s Mercury 0.206 orbital eccentricity. Write the equation of an Venus 0.007 ellipse centered at the origin that can be used as a model of Earth’s orbit. Earth 0.017 Mars 0.093 SOLUTION Use the eccentricity to find the Jupiter 0.048 relationship between a and b. Saturn 0.056 b2 e = 0.017 = 1 - _ Uranus 0.047 √ a2 Neptune 0.009 b2 0.0003 = 1 - _ a2 b2 _ = 0.9997 a2 y b = 0.9998a 0.5 Math Reasoning A scale model of Earth’s orbit maintains this O x relationship between a and b. So, let a = 1 to get Analyze Why does the -0.5 0.5 Earth’s orbit look more this equation and graph. -0.5 circular than oval? x2 y2 _ + _ = 1 1 2 0.99982

Lesson Practice a. Write an equation in standard form for the 6 y (Ex 1) ellipse with center (0, 0). 4

Graph the following equations. Find the center, 2 vertices, co-vertices and the foci. Calculate the x eccentricity e. -6 -4 -2 2 46 b. 16x2 + 36y2 = 576 -2 (Ex 2) c. 9(x - 2)2 + 64(x - 5)2 = 576 -4 (Ex 3) d. Write the equation of an ellipse centered -6 (Ex 4) at the origin that can be used as a model of Mars’s orbits.

Lesson 98 687 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PPageage 668888 66/10/08/10/08 9:15:169:15:16 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

Practice Distributed and Integrated

⎡ ⎤ - 5 2 11 *1. Graphing Calculator Calculate the determinant of 4 - 1 0 using a graphing (14) ⎢  calculator. ⎣ 19 1 1⎦

*2. Generalize How many ellipses have an eccentricity of 0.5 and cross the minor axis (98) at (0, 2)?

_x2 _y2 *3. Optics An elliptical mirror is modeled by the equation 2 + 2 = 1. The property (98) 15 10 of an elliptical mirror is that if you shine a beam of light from a focal point to the mirror, the light is reflected toward the other focal point. What is the length of the two light beams?

4. Find the missing side lengths of ABC. A = 73º; a = 18; b = 11 (71)

*5. Write the equation of the ellipse given by 2x2 + 3y2 = 6 in standard form. (98) 6. Packaging The manager of a company that sells pasta is looking into a new (66) container shape of a rectangular prism whose base is square and whose height is 5 inches greater than the length of one side of the base. If the volume is 28 cubic inches, what is the length of one side of the base? What is the height?

7. Geometry A rectangle has length 4c and width d. Another rectangle has length (2) 16 and width (2 + 5c). Write a simplified expression of their combined area.

*8. Population The estimated population of the District of Columbia from 2005 to 2006 (97) decreased by about -0.089%. Assuming that the percent of decrease stays the same, the populations each year form a geometric sequence. Show how to use the formula for the nth term of a geometric sequence to find the population of DC in 2012 given that the estimated population in 2005 was 582,049. (Hint: Call 2005 year 1.)

9. The Ehrenberg Relation The Ehrenberg relation log(w) = 0.8h + 0.4 ± 0.04 shows the (87) relationship between height h and weight w in young children ages 5 through 13. Solve for h and determine the height, in meters, of a child who weighs 40 kilograms.

10. Write Explain how to graph 3x + 5y = -35 using slope and intercept. Be sure to (13) show each step.

*11. Error Analysis Explain and correct the error a student made in finding the common (97) ratio of the geometric sequence 270, 90, 30, 10, …

r = _30 = 3 10 12. Given h(x) = 6 sin(8x - 9), identify the amplitude and period. (82) 13. Find the roots of the polynomial function. (76) f (x) = (x + 8)(3x2 - 6) - (x + 8)(11x - 4)

*14. Justify Suppose that for polynomials P1(x) and P2(x) there are constants a and (95) b such that P1(a) ≠ 0 and P2(b) ≠ 0. Is it correct to conclude that b is not a root

of P1(x) and a is not a root of P2(x)? Explain. 688 Saxon Algebra 2 SSM_A2_NLB_SBK_L098.inddM_A2_NLB_SBK_L098.indd PPageage 668989 66/11/08/11/08 5:34:205:34:20 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L098

15. Acoustics A guitar string is strummed twice, which creates two waves along (86) the string. The first wave is reflected back and meets the second wave. If the waves are modeled by the equations below, what happens when the waves meet? (Hint: When one wave’s crest coincides with another wave’s trough, they tend to cancel each other out. This called destructive interference.)

_π _π y = cos (x - 2 ) y = cos( x + 2 )

1 16. Error Analysis A student attempted to solve the formula V = _ πr3 for r and wrote (88) 3 3 _V r = √ 3π . What is the error? Solve the formula for r correctly.

Simplify. -2 -3 0 2 __(x yp) (x yp) _3x2y _2 x -2 x4 _5xy2 17. -2 18. - + - (3) (2x2 ) (3) xx y -1 x2 xy 19. Multi-Step A rectangular window has dimensions of 9 feet by 12 feet. The homeowner x ft (89) wants to increase the window size by lengthening two sides by the same amount as shown. The area of the window is to remain less than 300 square feet. 12 ft a. Write an inequality to find the range of the amounts that can be added to the two sides. b. Solve the inequality. 9 ft x ft

sin(x) 20. Model For what values of θ is y = _ the equivalent of y = tan(θ)? (90) cos(x - θ)

Evaluate each piecewise function for x = 3 and x = 0. ⎧_1 , if x < 1 ⎧ 2 2 x - 5, if x ≤ 0 21. f (x) = ⎨ 22. f (x) = ⎨x , if 1 ≤ x < 10 2 (79) ⎩x , if 0 < x < 3 (79) ⎩ -x, if x ≥ 10

23. Write a quadratic equation whose roots are 5i and -5i. (83)

Identify all the real roots. 24. x3 + 10x2 + 17x = 28 25. x3 - 343 = 0 (85) (85) _x2 + 5x - 6 *26. Multiple Choice When solving 2 > 0 by using a sign table, for how many (94) x - 12x + 32 intervals will a test point need to be chosen? A 3 B 4 C 5 D 6

Simplify. 1 1 _ - _4 _ - _1 _2 15 __x + 1 x - 1 27. 28. x (48) _5 _1 (48) _ + 2 6 9 x - 1 29. Write How and why is the LCD used to solve a rational equation? (84) *30. Probability A point is randomly chosen inside the graph of the polar equation r = 3. (96) What is the probability that the chosen point is at least 1 unit away from the pole? Lesson 98 689 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PPageage 669090 66/10/08/10/08 9:32:239:32:23 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L099Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L099

LESSON Using Vectors 99

Warm Up 1. Vocabulary For the complex number a + bi, the a term is known as the (62) . 2. True or False: A 2 × 3 matrix can be multiplied by another 2 × 3 matrix. (9) 3. What is the distance between the points (3, 7) and (-2, 3)? (41)

y For vectorz : New Concepts A vector is a quantity that has 8 magnitude, orientation, and direction. ⎪z ⎥ → magnitude 6 θ → orientation There are several ways of expressing vector quantities, including complex 4 z

numbers and matrices. Vectors can be 2 Vector graphed on Cartesian coordinate systems O θ x as well as complex coordinate systems. 2 4 6 8

Example 1 Graphing Vectors

a. Graph vector z whose endpoints are the origin and (5, 7) on the Cartesian and complex planes. Express the Cartesian version as a coordinate matrix and the complex version as a complex number. Find the orientation of the vector. SOLUTION Cartesian Coordinate System Complex Coordinate System Math Reasoning y Verify How do you 8 8i (5, 7) z = 5 know that the angle of 7 orientation is 54.46°? 6 6i z = 5 + 7i 4 4i

2 2i O θ = 54.46° x O 2 4 6 8 2 4 6 8

b. Graph vector z whose endpoints are the origin and (-15, -3) on the Cartesian and complex planes. Express the Cartesian version as a coordinate matrix and the complex version as a complex number. Find the orientation of the vector. SOLUTION Cartesian Coordinate System Complex Coordinate System Math Reasoning y y 1 1i Verify How do you θ = 191.31° know that the angle of x x orientation is 191.31°? -18 -12 -6 -18 -12 -6 -1 -1i z = 15 3 -2 -2i

-3 -3i z = 15 3i

690 Saxon Algebra 2 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PagePage 691691 6/4/086/4/08 3:46:283:46:28 PMPM useruser //Users/user/Desktop/Anil_04-06-08/currentUsers/user/Desktop/Anil_04-06-08/current

Hint Vectors can be added and subtracted.

When adding terms in Vector Addition Vector Subtraction matrices, the terms are added according to their ⎤ x + x ⎤ x - x ⎡x 1 ⎡ x2 ⎤ ⎡ 1 2 ⎤ ⎡x 1 ⎡ x2 ⎤ ⎡ 1 2 ⎤ positions. Vector addition ⎢  + ⎢  = ⎢  ⎢  - ⎢  = ⎢  and subtraction is similar ⎣y1⎦ ⎣y2 ⎦ ⎣ y1 + y2 ⎦ ⎣y1⎦ ⎣y2 ⎦ ⎣ y1 - y2 ⎦ to matrix addition and subtraction. (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) - (c + di) = (a - c) + (b - d)i Vector Addition Vector Subtraction y y

x

O x

Example 2 Adding and Subtracting Vectors

a. Find the vector sum A + B using matrix and complex addition. y 4 (3, 4) (1, 3) 3 A 2

1 (2, 1) O B x 1 2 3 4

SOLUTION ⎡ 1⎤ ⎡ 2⎤ ⎡ 3⎤ ⎢  + ⎢  = ⎢  (1 + 3i) + (2 + i) = 3 + 4i ⎣ 3⎦ ⎣ 1⎦ ⎣ 4 ⎦

b. Find the vector difference A - B using matrix and complex subtraction.

O y x -4 -2 2 4 -1 -B A -2 (-3, -2) (4, -2) -3

-4 (1, -4)

SOLUTION ⎡ 4⎤ ⎡ 3⎤ ⎡ 1⎤ ⎢  - ⎢  = ⎢  ( 4 - 2i) - (3 + 2i) = 1 - 4i ⎣ -2 ⎦ ⎣ 2⎦ ⎣ -4⎦

Online Connection Vectors can be multiplied by a process known as finding the dot product. The www.SaxonMathResources.com dot product is a scalar quantity, not a vector.

Lesson 99 691 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PPageage 669292 66/10/08/10/08 9:32:409:32:40 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L099Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L099

The dot product can be found by using the coordinate vectors. ⎡ ⎤ x2 [x1y1] × ⎢  = x1x2 + y1y2 ⎣y2⎦

Example 3 The Dot Product of Two Vectors

y Find the dot product A · B. 8 (3, 7) SOLUTION 6 A ⎡ 5⎤ [3 7] × ⎢  = 15 + 21 = 36 4 (5, 3) ⎣3⎦ 2 B O x 1 2 3 4

The magnitude of a vector, represented by A ⎢, in standard position can be determined by using the distance formula with the endpoints of the vector,

the origin (0, 0) and (x1, y1). 2 2 2 2 A⎢ = √( x1 - 0) + (y1 - 0) = √( x1) + (y1) The angle between two nonzero vectors can be found using the dot product of the two vectors and the magnitude of each vector.

x1x2 + y1y2 cos θ = ___ 2 2 2 2 √( x1) + (y1) √( x2) + (y2)

x1x2 + y1y2 θ = arccos ___ 2 2 2 2 √( x1) + (y1) √( x2) + (y2)

Example 4 Finding the Angle between Two Vectors y Find the angle between the vectors A and B . 8 (5, 7) 6 SOLUTION A 4 Use the coordinates to find the angle. 2 (3, 2) x1x2 + y1y2 θ = arccos __ O B x 2 2 2 2 ( √x 1 + y 1 √ x 2 + y 2 ) 1 2 3 4 5 · 3 + 7 · 2 29 θ = arccos __ = arccos __ ( √5 2 + 72 √3 2 + 22 ) ( √25 + 49 √9 + 4 ) = arccos(0.935) = 20.77°

y If you know the magnitude and orientation of a 8 vector, you can derive the horizontal and vertical 6 components by using the properties of a right triangle. _x 4 cos θ = x = ⎪A ⎥ cos θ A ⎪A ⎥ y 2 y sin θ = _ y = ⎪ A ⎥ sin θ O θ x ⎪A ⎥ 2 4 6 8

692 Saxon Algebra 2 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PagePage 693693 6/4/086/4/08 3:47:383:47:38 PMPM useruser //Users/user/Desktop/Anil_04-06-08/currentUsers/user/Desktop/Anil_04-06-08/current

Example 5 Application: Navigation An airplane traveling north at 500 mph hits a 60 mph θ = 30° headwind blowing 30° south of due west. What is the 60 500 actual speed and direction of the plane? SOLUTION Step 1: Find the horizontal and vertical components. ⎡ 0 ⎤ ⎡ -60 cos (30°)⎤ ⎡ -51.96⎤ ⎢  + ⎢  = ⎢  ⎣ 500⎦ ⎣-60 sin (30°) ⎦ ⎣ 470 ⎦ Step 2: Find the magnitude and direction. Magnitude Direction

2 2 _-51.96 √(- 51.96) + 470 = 472.86 θ = arccos( 472.86 ) = 96.31° NNW The plane is heading 96.31° NNW at 472.86 mph.

Lesson Practice a. Express a vector whose endpoint is at (10, 5) as a coordinate matrix and (Ex 1) a complex number. b. Express a vector whose endpoint is at (-12, -36) as a coordinate (Ex 1) matrix and a complex number.

⎡ 5⎤ ⎡ -6⎤ c. Add the vectors ⎢  and ⎢ . (Ex 2) ⎣4⎦ ⎣ 4 ⎦

⎡ -10⎤ ⎡ -3⎤ d. Subtract the vectors ⎢  and ⎢  . (Ex 2) ⎣ 7 ⎦ ⎣ 9 ⎦ ⎡ -10⎤ ⎡ -3⎤ e. Find the dot product between the vectors ⎢  and ⎢  . (Ex 3) ⎣ 7 ⎦ ⎣ 9 ⎦ ⎡ -8⎤ ⎡ 9 ⎤ f. Find the angle between the vectors ⎢  and ⎢  . (Ex 4) ⎣ 12 ⎦ ⎣-12⎦ g. An airplane traveling north at 600 mph hits a 30 mph headwind blowing (Ex 5) east at a 40° angle from north. What is the actual speed and direction of the plane?

Practice Distributed and Integrated

1. Error Analysis Explain the error a student made below. (70) 2 5 + √ x = 3 √ x = -2 ( √ x ) = (-2 ) 2 x = 4 The solution is 4.

*2. Generalize How many ellipses have an eccentricity of 0? Explain. (98)

2 3. Write Explain why ln e x +3 = x 2 + 3. (81) 4. Engines Two-stroke engines require a premix of gas and oil. The amount of gas (8) used varies directly with the amount of oil used. In some two-stroke engines, 5 gal of gas requires 20 oz of oil. Lesson 99 693 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PagePage 694694 6/4/086/4/08 3:47:463:47:46 PMPM useruser //Users/user/Desktop/Anil_04-06-08/currentUsers/user/Desktop/Anil_04-06-08/current

a. Write the proportion that can be used to find how many ounces of oil to use with 9 gal of gas. b. Solve the proportion in part a. c. Check the answer from part b with the proportion from part a.

5. Geometry Write a polynomial in standard form that represents the total area x (11) _1x formed by the rectangle and trapezoid. (Hint: The formula for the area of a 2 1 trapezoid is A = _ ( b + b )h). 2 1 2 x

x + 10

6. Probability A student has a spinner with four equal sections labeled 2, 4, 6, and 8. (68) If the student spins the spinner two times, find the probability that the first spin lands on a 4 and the sum of the results is greater than or equal to 10. ⎧ x - x 2 , if x ≤ 1   12, if 1 < x ≤ 2 7. Evaluate the piecewise function f (x) = ⎨ for x = -0.4, (79)  x = 0, and x = 6. - 6, if 2 < x < 4  ⎩ -x -8, if x ≥ 4

8. Error Analysis Explain and correct the error a student made in finding the z-score for (80) a data value of 15, taken from a normally distributed set of data, where the mean is 20 and the standard deviation is 4.

z = _20 - 15 = _5 = 1.25 4 4

9. In RST, ST = 17, RT = 20, and m∠T = 38°. Find m∠S. (77) 10. Write If two quintic polynomials are added, what are all the possible types of the (11) resulting polynomial (classified by degree)?

4 3 y 11. The graph of y = √ x + 1 + √ x + 1 is shown to the right. Use the axis of (75) 8 symmetry to draw the inverse function. 6 12. Seismology To locate the epicenter of an earthquake, a seismologist graphs the (91) 4 circles created from three seismographs in different locations and finds their point of intersection. a. The equations below were created from the readings of three seismographs O x where each unit represents one mile. Sketch the circles on the same coordinate -8 -4 4 8 plane. (x + 4)2 + (y - 4) 2 = 25 (x - 1 ) 2 + (y + 3) 2 = 49 (x - 5) 2 + (y - 4) 2 = 16 b. Estimate the location of the epicenter.

13. Prom A member of the prom committee polls 40 random seniors from (73) the senior class meeting to ask about possible locations for the prom. Describe the individuals in the population and in the sample.

694 Saxon Algebra 2 SSM_A2_NLB_SBK_L099.inddM_A2_NLB_SBK_L099.indd PagePage 695695 6/4/086/4/08 3:47:553:47:55 PMPM useruser //Users/user/Desktop/Anil_04-06-08/currentUsers/user/Desktop/Anil_04-06-08/current

14. National Parks Kolob Arch in Zion National Park is one of the longest arches in (83) the world with a span of about 288 feet and a height of about 105 feet. Write a quadratic function to approximate the arch.

I 15. Sound Intensity The loudness of a given sound is represented by d = 10log _ where (87) I o d is decibels, I is intensity of the sound, and I o is the intensity of a sound that can just be heard by humans. Determine the decibel level of a sound with an intensity 9 of (2.3 × 1 0 ) I o .

3 2 16. Use cross multiplying to show that _ = _ has no solution. (84) x + 5 x + 5 17. Write An English teacher needs to pick 5 students to present book reports to the (73) class. The teacher writes the names of all students in the class on pieces of paper, puts the pieces in a hat, and chooses 5 names without looking. Determine what type of sample is used and whether it is biased or unbiased. Explain your answer.

*18. Analyze Can a quadratic inequality have no critical values? Explain. (89) *19. Coordinate Geometry The points (1, -5), (2, -11), (3, -17), (4, -23), and (5, -29) (92) represent the first five terms of an arithmetic sequence. What is the common difference? What is the y-coordinate when the x-coordinate is 12?

*20. Graphing Calculator Use a graphing calculator to find the inverse of the matrix, if it (32) ⎡ ⎤ 2 1 3 exists. H = 1 1 2 ⎢  1 4 6 ⎣ ⎦ ⎡ 6⎤ ⎡ 2⎤ *21. Find the dot product between the vectors ⎢  and ⎢  . (99) ⎣8⎦ ⎣7⎦

*22. Multiple Choice Which are polar coordinates for the point with Cartesian (96) coordinates (-3, -3 √3 ) ? A 3, _7π B 3, _4π C 6, _7π D 6, _4π ( 6 ) ( 3 ) ( 6 ) ( 3 ) *23. Write an equation in standard form for the ellipse with center (0, 0), vertex (6, 0), (98) and covertex (0, -5).

Solve.

24. 3 _2 x - 4 _1 x = 2 _1 25. 0.02(p - 2) = 0.03(2p - 6) (2) 5 10 4 (2)

*26. Use the Rational Root Theorem to find the roots of y = 2x3 - 3x2 - 8x + 12. (85)

Add. 2 2 x c 27. 3x y m + _4 28. _5 - 4 + _ (37) x (37) pm p 2 m

⎡ -3⎤ ⎡ 3⎤ *29. Multi-Step Find the magnitude of vectors ⎢  and ⎢ . Calculate the angle (99) ⎣ 7 ⎦ ⎣7⎦ between the two vectors. Then calculate the dot product.

30. Find the distance between (-3, 7) and (4, -2). (41)

Lesson 99 695 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PagePage 696696 6/12/086/12/08 5:43:285:43:28 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

LESSON Graphing Rational Functions I 100

1 Warm Up 1. Vocabulary In order for _ to be a rational expression, g(x) must (28) g(x) be a not equal to zero. x2 x 2. Write __4 - 11 - 20 in factored form. (23) 3x2 + 21x - 54

3. What values of x are not allowed for this expression? __1 (23) 7x2 + 3x - 4

New Concepts A rational function is of the form g(x) f(x) = _ for polynomials g(x) and h(x). The domain of f(x) is all real h(x) numbers except those for which h(x) = 0. A real number a that makes the denominator of a rational function equal to zero is not in the domain of the function. The graph of the function is discontinuous at x = a. y 4

2 O 4 -2 Vertical asymptote: x = a -4 such that h(a) = 0

Example 1 Finding Points of Discontinuity

Find any points of discontinuity for each rational function. x + 1 a. y = _ x2 + 5x +6 SOLUTION The function is undefined when the denominator equals zero. Set the denominator equal to zero and solve. x 2 + 5x + 6 = 0 (x + 2)(x + 3) = 0 (x + 2) = 0 (x + 3) = 0 x =-2 x=-3 Online Connection www.SaxonMathResources.com The points of discontinuity are -2 and -3.

696 Saxon Algebra 2 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PPageage 669797 66/10/08/10/08 11:55:0911:55:09 PMPM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

x+5 b. y = _ x2+6 SOLUTION The function is undefined when the denominator equals zero. Set the denominator equal to zero and solve. x2+6 = 0 x2 = -6 x = ± √- 6 = ±i √6 Since i √6 is not a real number, there is no point of discontinuity.

A vertical asymptote or a hole in the graph occurs at a point of discontinuity. If there are no common factors in the numerator and denominator, there are vertical asymptotes at the points of discontinuity. If there are common factors in the numerator and denominator, there are either holes or vertical asymptotes at the points of discontinuity.

Example 2 Finding Vertical Asymptotes and Holes In each rational function, describe the vertical asymptotes and holes for the graph. (x + 1) a. y = __ (x + 4)(x - 2) SOLUTION The points of discontinuity are -4 and 2 and since these are no common factors of the numerator and denominator, x = -4 and x = 2 are vertical asymptotes. Hint (x + 5)(x + 6) b. y = __ When the factors in (x + 6)(x + 5)(x + 6) the numerator and denominator are the SOLUTION same and thus cancel, a hole will occur at the Since there are common factors in the numerator and denominator, the x-value that causes the _1 graph of this rational function is the same as y = x + 6 , except it has a hole factor to equal zero. at x = -5. The vertical asymptote is at x = -6.

The graph of a rational function has at most one horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator the graph has a horizontal asymptote at y = 0. If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote y = 0. If the degrees of the numerator and denominator are equal, _a then the graph has a horizontal asymptote at y = b , where a is the coefficient of the term with the highest degree in the numerator and b is the coefficient of the term with the highest degree in the denominator.

Lesson 100 697 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PPageage 669898 66/10/08/10/08 10:08:0510:08:05 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

Example 3 Finding Horizontal Asymptotes Find the horizontal asymptotes of each rational function. x2 a. y = _3 + 2 x2 - 5 SOLUTION Since the degree of the numerator and denominator are equal, the equation _3 of the vertical asymptotes is y = 1 = 3. 2x2 + 3x +7 b. y = __ x - 5 SOLUTION Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Example 4 Graphing Using Asymptotes

Math Reasoning Graph the following function. Identify any asymptotes. 2 Verify Show that as x a. y = _ gets infinitely large, there x - 1 is a horizontal asymptote at y = 0. SOLUTION Determine any points of discontinuity. y 4 x - 1 = 0 2 x = 0 O There is a point of discontinuity 2 4 Vertical at x = 1. Since the numerator and -2 denominator have no common factors, asymptote: x = 1 x = 1 is a vertical asymptote. -4 Since the degree of the denominator is greater than the degree of the numerator, the graph has a horizontal asymptote at y = 0. Calculate the values of y for values of x near the asymptotes. Plot these points and sketch the graph.

x2 + 2x - 8 b. y = __ Math Reasoning x3 - 4x SOLUTION Generalize Identify the holes and vertical x2 + 2x - 8 (x - 2)(x + 4) (x - 2)(x + 4) asymptotes for rational y = __ = __ = __ functions of the form x3 - 4x x(x2 - 4) x(x + 2)(x - 2) x - a y = _ . x2 - a2 (x + 4) = _ x(x + 2)

A common factor divides out of both the numerator and denominator, leaving vertical asymptotes x = 0 and x = -2. However, x = 2, while not an asymptote, is still not part of the function and is graphed as a “hole.”

698 Saxon Algebra 2 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PagePage 699699 6/12/086/12/08 9:24:009:24:00 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

The x-intercepts of a rational function occur when the value of the function is zero for any value in the domain. The x-intercept is -4. Calculate the values of y for values of x near the asymptotes. Plot these points and the x-intercepts. Sketch the graph.

y 4

2 O -4 2 4 -2

Lesson Practice Find any points of discontinuity for each rational function. (Ex 1) x + 2 (2x + 7)(x + 3) a. y = _ b. y = __ x2 - 4x -5 (x + 3) In each rational function, describe the vertical asymptotes and holes for the graph. (Ex 2) (x + 7) (x + 5)(x + 6) c. y = __ d. y = __ (x + 3)(x - 5) (x + 6)(x + 5)(x + 6) Find the horizontal asymptotes of each rational function. (Ex 3) x2 e. y = _6 + 3 2x2 + 4 2x2 + 3x + 7 f. y = __ x - 5

Graph the following functions. Identify any asymptotes or holes. (Ex 4) g. y = _4 x - 2 h. y = _2 x3 - x Graph the following functions. Identify any asymptotes, holes, and x-intercepts. (Ex 5) x2 x i. y = __- 2 - 15 x2 + 5x + 6 x2 x j. y = __- 2 - 15 x3 - 9x

Lesson 100 699 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PPageage 770000 66/3/08/3/08 4:20:164:20:16 PMPM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

Practice Distributed and Integrated

Solve. 5 1. x = 4 + _ 2. -x2 - 2x - 1 > 2 (84) x (84) 12 *3. Error Analysis Two students calculated the angle between vectors A = and (99) 9 17 B = but their results were different. Which student made the mistake? 8

Student A Student B (12 · 17) + (9 · 8) (12 · 9) + (17 · 8) θ = arccos __ θ = arccos __ ( √12 2 + 92 √17 2 + 82 ) ( √12 2 + 92 √17 2 + 82 ) = 11.67° = 30.03°

4. Find the equation of the line parallel to y = 7x + 6 that crosses (6, 10). (36) *5. Phone Service A planned cell phone tower will service an area with a 13-mile (96) radius. It is to be located 5 miles east and 12 miles south of a radio tower. Write a polar equation of the boundary of the region that will be serviced. Place the location of the radio tower at the pole. Graph the equation on a calculator. _180°(n - 2) 6. Multi-Step The formula θ = n gives the measure of each interior angle of (56) an n-sided regular polygon. a. Use the formula to find the measure of an interior angle of a regular decagon. b. Find the reference angle for the interior angle. x + 20 *7. Geometry A rectangular prism has the dimensions shown. The volume V(x) is (95) x - 4 the product of the linear dimensions. Use the Remainder Theorem to evaluate V(20). x + 7

Write a piecewise function rule for each graph. 8. y 9. y (79) (79) 8 8 4 4 O x O x -8 -4 4 8 -8 -4 4 8 -4 -4 -8

*10. Multiple Choice Which of the following rational functions includes a vertical asymptote (100) at x = 7? x x A y = _- 7 B y = __- 7 x2 - 49 x2 + 12x + 35 x2 x2 C y = __ - 16 D y = __ - 16 x2 + 11x + 28 x2 - 11x + 28

700 Saxon Algebra 2 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PPageage 770101 66/3/08/3/08 4:20:244:20:24 PMPM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

2 11. Analyze Use the table of values for y = lo g 2 x to answer the questions. (87) x -4 -3 -2 -1 01234 y 4 3.17 2 0 - 0 2 3.17 4 a. As the negative x values increase to 0, what happens to the y values? b. As the positive x values increase, what happens to the y values?

12. Justify Can the difference of two polynomials have a greater degree than the sum (11) of the same two polynomials? Justify your answer with an example.

*13. Graphing Calculator Use a graphing calculator to evaluate y = 2x + 3 for the given (22) domain: -2, -1, 0, 1, 2. Identify the range of the function and classify the function.

14. Parking An airport parking garage costs $38 per day for the first week. After (79) that, the cost decreases to $35 per day. Write a piecewise function for the cost of parking x days.

15. Analyze Suppose that for polynomials P1(x) and P2(x) there is a constant a such that (95) P1(a) = P2(a) = 0. What can you conclude about the graphs of P1(x) and P2(x)?

16. Research In a study of 413 men and 335 women, the mean height of men was (80) 174.1 centimeters with a standard deviation of 7 centimeters. The mean height of women was 162.3 centimeters with a standard deviation of 6.2 centimeters. Assume both distributions are normal. Compare the actual height of a man with a z-score of 2 to the actual height of a woman whose z-score is 2.

17. Data Analysis The number of days it takes to complete a year for different planets is (82) shown in the table. Let Earth’s yearly motion be modeled by y = sin(0.017x). How would you model Jupiter’s motion? Hint: The days in a year corresponds to the period of the function.

Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Days in 88.03 224.63 365.25 686.67 4331.87 10,760.27 30,681.00 60,193.20 a Year

*18. Error Analysis Which student made an error in determining intervals to check for (94) x - 2 solutions and nonsolutions of _ ≥ 0? What is the error? (x + 7)(x - 5)

Student A Student B x ≤ -7 x < -7 -7 ≤ x ≤ 2 -7 < x ≤ 2 2 ≤ x ≤ 5 2 ≤ x < 5 x ≥ 5 x > 5

19. Analyze Identify which property of real numbers is being demonstrated by (1) (4 · 6) · 5 = 4 ·(6 · 5) = 4 · 30 = 120. Explain why this property might be helpful in solving this problem without a calculator.

Lesson 100 701 SSM_A2_NLB_SBK_L100.inddM_A2_NLB_SBK_L100.indd PPageage 770202 66/3/08/3/08 4:20:384:20:38 PMPM user-s191user-s191 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Lessons/SM_A2_SBK_L100

20. Multi-Step The perimeter of a rectangular garden will be 100 feet. Write an (89) inequality to find the widths that will give an area of at least 450 square feet. Solve the inequality and round to the nearest tenth.

sin(x) 21. For what values of θ is y = _ no longer a periodic function? (90) cos(x - θ) 22. Physics Every millisecond a radio beacon emits a signal. If the magnitude of the (82) signal is 100, write the periodic function that models the signal using the form y = acos(bx).

Simplify. 23. 2 √3 ( 5 √3 - 2 √6 ) 24. 2 √3 · 3 √6 · 5 √12 (40) (40) 25. Write A student wants to graph a circle whose equation is (x - 5)2 + (y - 5)2 = 2 (91) on a graphing calculator. How would you explain to the student how this can be done?

26. Physics In a parallel circuit, the total resistance, RT, is found using (100) _1 _1 _1 RT = + + ... . Suppose that a parallel circuit has two resistors R1 R2 R3 of magnitude R and 2R + 1. Solve for RT.

Divide. 27. x4 + 5x3 - 6x2 by (x - 1) 28. x4 - 20x3 + 123x2 - 216x by (x - 3) (38) (38)

⎤ ⎤ ⎤ ⎤ ⎡ 10 ⎡ 4 ⎡ 10 ⎡ 3 29. Add vectors ⎢ and ⎢ . *30. Subtract vectors ⎢ and ⎢ . (99) ⎣ 6 ⎦ ⎣5⎦ (99) ⎣ 4 ⎦ ⎣14⎦

702 Saxon Algebra 2 SSM_A2_NLB_SBK_Inv10.inddM_A2_NLB_SBK_Inv10.indd PPageage 703703 66/11/08/11/08 2:04:432:04:43 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10

INVESTIGATION 10 Graphing Polar Models

An equation in polar coordinates uses the variables r and θ. Polar Equations for Circles A circle can be graphed using the polar equations: r = a sin θ r = a cos θ _π _π 2 2 _3π _π _3π _π 4 4 4 4 r = sin θ r = cos θ

π 0 π 0 -1 1 -1 1 -1 -1 _5π _7π _5π _7π 4 4 4 4 _3π _3π 2 2

1. How do the graphs of r = sin θ and r = cos θ compare to the unit circle?

Graphing 2. Graph r = sin θ and then graph r = a sin θ for three values of a where Calculator 0 < a < 1. How did the value of a change the graph from the parent To graph polar equations graph? with your graphing 3. Graph r = cos θ and then graph r = a cos θ for three values of a where calculator, press the key and select a > 1. How did the value of a change the graph from the parent graph? the POL option. Once in this mode, pressing the Polar Equations for Spirals key allows you to input an equation in One of the simplest polar equations that uses the variable θ is r = θ. r = f(θ) mode. Its graph is a spiral. _π 2 _3π _π 4 4 4

r = θ π 0 -4 -2 2 4 -2

_5π -4 _7π 4 4 _3π 2

4. Make a table of values for r = θ on a graphing calculator. How does the value of r change compared to changes in θ ? 5. Graph r = θ and then graph r = cθ for three values of c where 0 < c < 1. How did the value of c change the graph from the parent graph?

Online Connection 6. Graph r = θ and then graph r = cθ for three values of c where c > 1. www.SaxonMathResources.com How did the value of a change the graph from the parent graph?

Investigation 10 703 SSM_A2_NLB_SBK_Inv10.inddM_A2_NLB_SBK_Inv10.indd PagePage 704704 6/12/086/12/08 4:41:094:41:09 PMPM User-17User-17 //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10

Graphing For linear equations of the form r = aθ + b, the graph is also a spiral. _π Calculator 2 The graph on the right _3π _π 8 shows values of θ ranging 4 4 from 0 to 2π. With a r = θ + 2 graphing calculator, you can set θ to higher values. r = θ π 0 Press on the key -8 4 and set θmax to a greater multiple of π. _5π -8 _7π 4 4 _3π 2 7. Graphing Calculator Describe the role of a and b in the graph of r = aθ + b.

Math-Reasoning A more general form of the equation r = θ is r = a sin(bθ) + c. Below are Verify Use a graphing some of the interesting graphs that result from different values of a, b, calculator to show that and c. The graph of r = sin(θ) + 1 is a heart-shaped curve known as a the general form of cardioid. The graph of r = 3 sin(2θ) is called a rose curve. the of the equation, π π r = a(sin(θ) + 1), always _ _ 2 2 yields the graph of a _3π _π _3π _π cardioid. 4 4 4 3 4 1 r = 3sin(2θ) π 0 π 0 -1 1 -3 3 -1 r = sin(θ) + 1 _5π _7π _5π -3 _7π 4 4 4 4 _3π _3π 2 2 8. Graph 3 rose curves given by the general equation y = sin (bθ) where b is an even integer. How does the number of petals relate to the value of b? 9. Graph 3 rose curves given by the general equation y = sin (bθ) where b is an odd integer and not equal to 1. How does the number of petals relate to the value of b? 10. What would be the value of b in the equation of a rose curve with 20 petals? 11. How many petals are in a rose curve given by the equation y = sin (19θ)? 12. Graph the rose curve y = a sin(2θ) where a = 1. Graph 3 more rose curves by choosing the value of a to be greater than 1. How did the graph change? 13. Graph the rose curve y = a sin(2θ) where a = 1. Graph 3 more rose curves by choosing the value 0 < a < 1. How did the graph change?

The Polar Equations for Conic Sections The general form of the polar equation for conic sections is shown below. By varying the value of the constant e, the graph will be a circle, ellipse, parabola, or hyperbola. l r = __ 1 + e sin (θ) 704 Saxon Algebra 2 SSM_A2_NLB_SBK_Inv10.inddM_A2_NLB_SBK_Inv10.indd PPageage 705705 66/11/08/11/08 2:06:252:06:25 AMAM useruser //Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10Volumes/ju110/HCAC061/SM_A2_SBK_indd%0/SM_A2_NL_SBK_Inv/SM_A2_Inv_10

Caution Graphing Calculator The variable e is the eccentricity, or a measure of the The variable e used curvature of a conic section. For each of the following graphs, let the value in polar equations of l remain constant, while varying the value of e. determines the eccentricity of the conic 14. Choose a value for l and let e = 0. What kind of conic section is this section. It is not the graph? constant e, which equals approximately 2.718. 15. Choose a value for l. Choose values of e from the range 0 < e < 1. What kind of conic section is this graph? 16. Choose a value for l and let e = 1. What kind of conic section is this graph? The variable l represents the vertical distance from the conic section’s focus or center and the curve itself. _π 2 _3π _π 4 8 4 l = 3 r = __3 1 + 0.5cosθ π 0 -8 4 8

_5π -8 _7π 4 4 _3π 2 17. In the case of a circle, what does the value of l represent? Graphing Calculator Write and graph the equation of a conic section based on the requirements below. 18. A parabola with l = 3 19. A circle of radius 7 20. An ellipse with l = 10

Investigation Practice

a. Give a parameter change for the circle equation r = 2 sin θ that will increase the size of the circle and translate up.

b. Describe the change of the graph r = 3θ from the parent function r = θ.

c. Write an equation for a rose curve with 14 petals.

d. Write an equation for a rose curve with 9 petals.

e. Write an equation for a rose curve that has the same number of petals but lengthened in the equation y = 2 sin(3θ).

2 f. What kind of conic section is given by the equation r = _ ? 1 + 0.5 sin(θ) 4 g. What kind of conic section is given by the equation r = _ ? 1 + 1 sin(θ)

Investigation 10 705