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

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Graphing a Circle and a Polar Equation 13 Graphing Calculator Lab (Use with Lessons 91, 96) 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 66/10/08/10/08 112:18:482:18:48 AAMM eelhilhi //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 2 2 2 radicand and squaring both sides gives r = x + y . 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 2 2 2 r (0, 0) x x + y = r . -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 66/10/08/10/08 112:19:022:19:02 AAMM eelhilhi //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 y1 + y2 2 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.
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