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Name Period Date FINAL EXAM REVIEW Quadratic Functions For the following quadratic functions, state the form (standard, vertex, or intercept), complete the table, vertex, evaluate for the given value, and graph. f(x) = –3x(x – 2) 1. g(x) = –3x2 + 12x – 8 2. Form: Form: x y Vertex: x y f(4) Vertex: 3. State the vertex of the parabola defined by the equation y = 7x2 – 56x + 37 4. State the axis of symmetry of the parabola defined by the equation y = –3(x + 5)2 – 7 5. State the vertex of the parabola defined by the equation y = 2x2 – 14x + 12 g(–3) For the following quadratic functions, state the form (standard, vertex, or intercept), complete the table, vertex, evaluate for the given value, and graph. 1 6. f(x) = –2(x – 2)(x + 2) 7. g(x) = x(x + 6) 2 Form: Form: x y Vertex: x y f(–7) Vertex: g(6) 8. Does the graph of the equation f(x) = 2(x + 3)2 – 4 open up or down? How do you know? Factor the following completely: 9. y = 3x2 – 6x – 45 10. y = 2x2 – 15x + 27 11. y = 9x2 – 6x + 1 12. y = 2x2 – 8x – 44 13. f(x) = 6x2 – 17x + 3 14. f(x)= 4x2 – x – 3 15. f(x)= 9x2 – 4 Solving Quadratics Review Solve the following by square-rooting, or factoring. SHOW ALL WORK!! 16. 3(x + 7)2 – 5 = 67 1 17. 2( x + 4)2 – 3 = 75 2 18. 2x2 – 15x + 27 = 0 19. 9x2 – 6x + 18 = 17 20. 2x2 – 8x – 26 = 16 21. 2(x + 3)2 – 7 = 43 22. 2(x – 4)2 + 7 = 10 23. 4x2 – x + 9 = 12 24. x2 = 8x – 35 25. 2(4x – 3)2 + 1 = 65 26. 3x2 – 6x – 45 = 0 27. 28. 72 (2x – 3)2 = 0 29. 9x2 – 4 = 0 30. (x – 1)2 + 16 = 7 31. 3x2 – 11x – 4 = 0 1 (2x + 1)2 + 10 = 31 3 32. 2x2 – 15x + 27 = 0 Solve using the quadratic formula 32. x2 – 10x – 14 = 0 33. x2 – 12x = –28 34. 2x2 = –9x – 7 35. What does it mean to solve a quadratic equation? (When you find the solutions to an equation, you are really finding?) Use the graphs below to solve the following equations. 36. –2(x + 3)2 + 8 = 0 37. x(x – 6) = –10 38. x2 – 6x = 7 39. –2(x + 5)(x + 1) = 8 -10 -8 -6 -4 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 Solve the following by graphing. 40. –(x – 5)2 + 6 = 2 x y Solve using any method. 32. 2x2 + 12x = 4 34. 2x2 – 4x – 8 = -x2 + x 4 6 8 10 41. x2 – 6x + 10 = 5 x y 33. 3x2 – 14x – 5 = 0 Quadratic Applications Review 35. 36. 37. A circus clown tosses a juggling ball into the air. The ball leaves the clown’s hand 6 feet above the ground and has an initial vertical velocity of 45 feet per second a) Write a quadratic equation that models the height of the ball as a function of time since in was thrown. b) For how many seconds is the ball in the air before it hits the ground? c) When will the ball be 20 feet high? d) When will the ball be 40 feet high? A child tosses a penny into a well that is 50 meters deep. The penny leaves the child’s hand (half a meter high) with an initial vertical force of 5 meters per second. a) Write a quadratic equation that models the height of the ball as a function of time since in was thrown. b) How long will it take the penny to reach the bottom of the well? c) What is the maximum height of the penny? d) When will the penny be at ground level? An apple falls from a branch on a tree 30 meters above a man sleeping underneath. a) Write a quadratic equation that models the height h of the apple as a function of time t. b) When will the apple strike the man assuming that his head is at ground level? 38. 39. An engineering student is in an “egg dropping contest”. The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking the egg. a) Write a quadratic equation that models the height h of the egg container as a function of time t. b) How long will it take for the egg’s container to hit the ground? A golf ball follows a path modeled by y = –0.005x2 + 1.1x where x is the horizontal distance and y is the height (both in yards). Will the ball clear a sand trap in its path that extends from 190 to 210 yards out? YES or NO Explain your reasoning. 40. A rug is to cover two thirds of the floor area of a 12 foot by 16 foot room. The uncovered part of the floor is to form a strip of uniform width around the rug. Find the distance from the edge of the rug to the wall. 41. Your uncle wants your help designing his new patio. He currently has a rectangular patio that measures 8 by 12 feet. He wants to add a border of flagstone with width x feet of around the current patio. He wants the total area of the new patio to be twice as big as the old patio. What should the width of the border be? 42. For 1990 to 1997, the number of cellular phone subscribers s (in millions) in the United States can be approximated by the model s = 0.84t2 + 1.39t + 5.13 where t = 0 represents 1990. In which year did cellular phone companies have about 33.7 million subscribers? 43. A seagull launches a clam towards the ground to attempt to break the shell. It took approximately 1.54 seconds for the shell to hit the ground after the seagull launched it from 50 feet in the air. With what initial force did the seagull launch the clam? 44. A rectangular field is enclosed by 300m of fencing. Find the dimensions of the field that produce the greatest area. 45. A rectangular quilt is to be put on display in a museum, but the exact dimensions are unknown. A notebook found with the quilt documented that 30,000 square inches of fabric were used and 546 inches of ribbon were sewn around three edges of the quilt when it was constructed. (Note: the fourth side did not have ribbon because it would have been covered by pillows on a bed). Find the dimensions of the quilt. FINAL EXAM REVIEW Integer Exponents Use the properties of exponents to simply the expression. It may be helpful to show the steps you take in the simplifying process—make sure to keep your work organized! You will probably need to use a separate piece of paper. 1. –(3x)2 2. 5 4. 84 86 5. (76)(7–6) 8. 1 2 3 2 7. 5 38 10. 13. x0y–2 3 3. 6. 4 43 46 9. 56 5 3 11. 14. 2y 3 y5 4 3 2 3 4 2 2 3 2 3 2 3 2 2 1 12. (3x)2 15. 5x 2 y 2x 1y 3 16. 3 xy 9 x 3 y 4 19. 9 x 2 y 3 x 3 y 3 1 6 x y 22. (2 x )3 4x 5 3 2x 23. x 2 5 25. 4 x 3 y 3 z 4 6 x 2 y 4 z 2 26. (3x3y2)2 • (2xy3)2 28. 2x 3 y 2 2 3 x y 29. 2x 3 2 x 31. 9x 5 y 7 (2xy )2 2 3 • 2 2 x y 6 x y 32. yz 2 ( x 2 y )3 z 17. (3 x )2 6x5 3x 2 20. 1 2 16 8 x 3 4 x 18. 4 x 0 2x 3 y 3 xy 3 y 1 21. x 3 y 4 3 3x 24. 8x 8 y 2 z 4 2x 2 y 3 z 6 27. (2y 2 y ) 2 3 xy 4 30. 2 xy 2 y 4 4 xy • 1 3 1 2x y 3x y 4 2 2 2 2 FINAL EXAM REVIEW Polynomials Completely factor the polynomials. 8. x3 + 3x2 + 2x + 6 9. 2x3 – 6x2 + 10x – 30 10. 2x3 – 12x2 + 5x – 30 11. -3x3 + 12x2 – 2x + 8 12. 16x3 – 48x2 – x + 3 13. 9x3 + 18x2 – 4x – 8 14. x4 – 9 15. 2x4 – 200x2 16. x4 + 5x2 – 24 17. 8x4 – 18x2 18. x4 – 7x2 + 10 19. 2x4 + 16x2 + 24 20. –x4 –5x2 – 6 21. 81x4 – 256 Find all of the solutions to the following polynomial equations. 22. x4 – 10x2 + 9 = 0 23. x4 – 10x2 + 24 = 0 24. 2x5 – 12x3 = -16x 25. 3x4 + 3x3 = 6x2 + 6x 26. 9x3 + 18x2 = 4x + 8 27. 2x4 – 200x2 = 0 28. 3x3 = -12x2 + 4x + 16 29. 2x3 – 12x2 – 5x + 30 = 0 Sketch a graph of the polynomial function. (HINT: factor to find zeros!) 1. f(x) = x4 – 17x2 + 16 2. f(x) = x3 + 7x2 – 4x – 28 3. f(x) = 8x4 – 18x2 4. f(x) = x3 – 5x2 – 9x + 45 5. f(x) = 5x3 – 5x Sketch a graph of the polynomial. Then write an equation for you graph. (You can leave the function in factored form) 6. • Positive leading coefficient • Zeros at –1, 7, 2, and -5 7. • Right-end of function falls • Left-end of function rises • Zeros at –3, 1, 8, -4 and 6 6 7 Determine a suitable equation for the polynomial function graphed below. Leave your function in factored form. 30. f(x)=__________________________ 31. f(x)=__________________________ Write an equation in standard form given the following information. 32. Write an equation in standard form given the following information. 33. (-3, 0) (5, 0) (8, 0) (7, 40)