Review for Final s1

HONORS ALGEBRA II Name ______

REVIEW FOR FINAL EXAM PART ONE: POLYNOMIALS AND LAWS OF EXPONENTS

SIMPLIFY

1) 8y 3   y  5 2) 3 2a b  4 a  2 b

3 3)  x23 xy  4 y 2   2 x 2  3 xy 4) xy

4 5) 2xy3 5 x 2 y 2  6) 3a2 

2 7) 2cd2 3 c 2 d  8) 2x3 3 x 3  4 x  1

9) 5x 2 3 x  2 10) 3x y 3 x  y

2 11) 2a  7 12)  x4 x 2 2 3 x 2  4

3cd 4 -9 5 - 1 - 4 2 13) ( x y)(2 x y) 14) 2 -2 (3c d )

PART TWO: FACTORING, RATIONAL EXPRESSIONS, AND EQUATIONS

SIMPLIFY x x1 x2  8 3x 2 2 x  3 1.   4.  x4 x  4 x 2  16 2x 3 x x1 x2  x  6 2 3 2.  5. 2 2 x24 x 2  1 x4 x  4 x  4 2 2 4 4 1 1 x a x  a 6.  3.  2 2x 1 2 x 1 x1 x  1  

7. 6a2  a  15 3x2 17 x 5 10.   8. x2 8 x   15 2 12 4 x1 2 x  1 5  x2 2x 1 x 9.   11. 1  x5 x  2 x2  3 x  10 6 8 HONORS ALGEBRA II 2 12 1 12.   2x2 11 x  15 9  x 2 2 x 2  x  15 SET UP AND SOLVE THE FOLLOWING WORD PROBLEMS

13. If the length of one side of a square is increased by 6 cm, and the length of an adjacent side is doubled, the new rectangle that is formed has an area 13 cm2 greater than the area of the original square. Find the length of a side of the square.

14. A rectangular oil painting is 15 inches wide and 36 inches long. The painting is enclosed by a frame of uniform width. The area of the frame is 472 in2. Find the width of the frame.

15. A large pipe can fill a tank in 3 hours less time than a smaller pipe, and the two together can fill the tank in 2 hours. How long would it take each pipe to fill the tank?

16. If the base of a square is increased by 21 cm and its height is decreased by 2 cm, the area of the rectangle formed will be three times the area of the square. Find the length of each side of the square.

17. How much of an 80% nitric acid solution must be added to 175 g of a solution that is 12% nitric acid to produce a 60% nitric acid solution?

18. A stamping machine can produce a metal refrigerator part every 18 seconds. A newer model can produce the same part in 12 seconds. How long would it take the two machines to produce 50 parts working simultaneously?

GRAPH THE FOLLOWING

x2 +3 x + 2 3x2 - 19 x + 6 19. y = 21. y = x +1 x2 -5 x - 6

x - 4 x - 7 20. y = 22. y = x + 5 x2 - 49 HONORS ALGEBRA II

PART THREE: RADICALS

SIMPLIFY

5 2 1. 2 25a 15. 6 3  1 1 2. 3  3 2 64 9 16. 5 2 3. 75 2 12 3 2 4. 5 5 20 17. 32a c 3 2 4 5. 4 3 r8 s 10 18. 32 19. 6. 49x10 48   12 2 7. 12 20. 4 2i  8. 2 20  2  45 21. 9. 3 4i  7  i  16 2 4  i 10. 4 7i 22.   3 4i 10 40 11. 23.  12 27  10 1 1 5 2i 24. i5 i 6   12. i7 i 8 3  i 25. 3 64 13. 28 2 98  63 26. 3 81x4 y  6 14.  5 3 5  3 

27. x2 x  5  4 28. 2x 7  x  3  1 29. 2x  1  3  0 30. 2 4x  3  x 31. x2 12  2 x

PART FOUR: QUADRATIC EQUATIONS

1. Solve by completing the square:  x2 8 x   6  x2 12 x   27  2x2 - 10 x = - 5

2. Solve 3x2  2 x  2  0 by using the quadratic formula. HONORS ALGEBRA II

Solve over using any method.

2 2 3.  x 2  25 10. 3x 6 x  4  0 2 4. 16x2  8 x  1  0 11.  x 3  12  0 5. 8x2  2 x  15 12. x2 9  3 x 6. x2 2 x  5  0 x2 x 1 13.    0 7. 2x x 2  3 1  x  3 2 6 2 8. x2  x 5  1  0 14. x6 x  4  0 2 9. x2 6 x  2 15. 3x 2 x  4  0 16. 5x2  12 x  8  0

For each of the following, calculate the value of the discriminant, and use this to determine the type of roots the equation has.

17. x2 5 x  8  0 18. x2 3 x  7 19. 2x2  3 x  0 20. 2x2  5 x  3  0

Graph completely: Identify x and y-intercept(s), the vertex, and an equation for the axis of symmetry.

18. y  x2  3 2 19. x2 y  2  1 2 20. y 2 x  4  5 21. y x2 4 x  5

22. If the vertex of a parabola is ( 5, -2 ), and ( 1, 3 ) is a point on the curve, find an equation for the parabola.

For each of the following, calculate the value of the discriminant, and use this to determine the type of roots the equation has.

21. x2 5 x  8  0 22. x2 3 x  7 23. 2x2  3 x  0 24. 2x2  5 x  3  0 25. 2x2  4 x  2  0 HONORS ALGEBRA II 26. A rectangle has a perimeter of 50 ft. What dimensions will maximize the area of this rectangle and what is the maximum area?

27. Find two real numbers with sum 14 and product as great as possible. Find both the numbers and the maximum product.

28. A potato farmer has 400 bushels of potatoes that she can now sell for $3.20 per bushel. For every week she waits, the price per bushel will drop by $0.10 but she will harvest 20 more bushels. How many weeks should she wait in order to maximize her income? What will her maximum income be?

29. Consider all pairs of positive integers whose sum is 20. What is the smallest value for the sum of their squares? What pair of integers produces this value?

PART FOUR: THE CONIC SECTIONS

Sketch the graphs of the following conic sections. Provide all “critical” points (i.e: center, vertices, foci)

1. y x2 4 x  3 2. 4x2 y 2  36 3. y x2 4 x  5 1 4. y  x2  2 x 3 5. x2 y 2  9 6. x2 y 2 6 x  3 y  9 2 2 7. 16x 1  9 y  1  144

8. Write an equation of the form x2 y 2  ax  by  c  0 for the circle with center ( 0, -4 ) and radius 5.

9. Find the coordinates for the midpoint and the length of the line segment whose endpoints are ( 3, -2 ) and ( 4, -1 ).

10. Write an equation for the circle centered at ( 1, 4 ) and with a radius of 3.

11. Write an equation for the circle centered at ( -9, 3 ) and with a radius of 2 3 .

12. Write an equation for the ellipse whose foci lie at ( 5, 0 ) and ( -5, 0 ) and whose vertices lie at ( 9, 0 ) and ( -9, 0 ).

13. Write an equation for the hyperbola whose foci lie at ( 7, 0 ) and ( -7, 0 ) and whose vertices lie at ( 5, 0 ) and ( -5, 0 ).

14. Find the equation of a parabola with focus at (4, 5) and directrix y = 9. HONORS ALGEBRA II

Chapter 10 (Quadratic Relations and Systems)

Determine an equation in x and y that defines the given variation and contains the given ordered pair:

1. y varies inversely as x; (3, 9)

3  2. y varies inversely as x; 10,  5 

3. If z varies jointly as x and y and z = 60 when x = 3 and y = 4; find z when x = 5 and y = 10

4. If z varies jointly as x and y and inversely as w, and z = -3 when x = 3, y = 4, and w = 12; find y when z = -12, x = 5, and w = 10

Find the solution of each system over

1. y = x2 + 4 2. x2 – y2 = 8 y = 5x x – 3y = 0

3. x2 – y2 = 16 4. x2 + y2 = 6 3x + y = 0 x + y = 2 3

5. x2 + 4y2 = 17 6. y2 – 3x2 = 23 3x2 – y2 = -1 x2 – 16y2 = 8

7. 2x2 + y2 = 33 x2 + 3y2 = 79

8. An isosceles triangle has a perimeter of 50 cm. The sum of the length of the base of the triangle and the height of the triangle is 31 cm. Find the area of the triangle.

9. Find the coordinates of all points that are 10 units from the origin and 17 units from the point (0, -9). HONORS ALGEBRA II

Chapter 11 (Exponents and Logarithms)

Evaluate:

 2  3 1. 7 2  2. 2 12 

2 3 2  3 3 2 3  3 3. 5g 5 4. 10 10

1 2 5. 92 6. 325

1  3  125  3 7. 16 4 8.   64 

Write in simplified radical form:

1 1. 4 32 y 6 2. 6 27r24

2 12 49  3. 27 4. 4   144 

Solve the following:

1. 16x = 2 2. 4x = 2

3. 10x = 1000 4. 53x – 1 = 25x + 4

x 3 2x – 2 3 – x 2x + 4 1  5. 125 = 25 6. 10 =   100 

3 4 6. Are y x  6 and y x  8 inverses? Show. 4 3

1 7. Find the inverse of y  x  5 4 HONORS ALGEBRA II

Evaluate the following:

1. log10.0001 2. log71

log1 128 3. 4. log1515 2

log 36 6 5. log25125 6. 6

7 9 7. log39 8. log525

9. log64 + 2log63 10. 2(log220 – log25)

1 1 11. log 16 – 2log 10 12. log2 27 2 5 5 23

13. log511 14. log7.385.9

Solve for x:

2 1. log 16 = ½ 2. log x =  x 1000 3

3. logx 7 = -2 4. log1001000 = x

5. logx125 = 6 6. log328 = x

2 7. log5(2x + 5) = 3 8. log3(x +17) = 4

1 1 9. log x = log 8 + log 81 10. log x2 = log 8 + log 10 – log 5 10 3 10 2 10 3 3 3 3

11. log3x – log34 = 2log35 12. log6x + log6(x + 5) = 2

2 -x 13. log32x – log3(5x – 9) = 1 14. 7.4 = 18.6

x 3x 15. 54 = 19 16. 1288  33.7

x 2x 17. .76(5 ) = 29.3 18. 4.1 8.2 2  132

19. 63x – 1 = 28x 20.42x – 1 = 173x – 1 HONORS ALGEBRA II

21. A bacteria culture is found to double in size every 24 minutes. How many minutes would it take for a culture of 320,000 bacteria to grow to 761,000 bacteria?

22. Forty years ago the population of a certain town was 2000. It is now 12,000. Assuming exponential growth, in how many years will the population be 45,000?

23. Find the value of an investment of $6000 after 1.5 years if the interest is compounded continuously at 8%.

24. How many years ago was $5000 invested in an account paying 8% annual interest compounded quarterly, if the amount presently in the account is $11,500?

Express in logarithmic form: Express in exponential form:

1. e-2 = .135 1. ln .5 = -.693

2. e 1.649 2. ln .01 = -4.605

Simplify:

1 1. ln 9 ln12  2ln 3 2. ln 6 + ln 30 – (ln 5 + 3 ln 2) 2

1 2 ln 7 ln3 3. e 4. e 2