QUADRATIC FUNCTIONS TEST REVIEW NAME: SECTION 1: FACTORING Factor each expression completely. 1. 3x 2 48 2. 25p 2 16p 3. 6x 2 13x 5 4. 9x 2 30x + 25 5. 4x 2 + 81 6. 6x 2 14x + 4 7. 4x 2 + 20x 24 8. 4x 2 + 20x + 25 9. 2x 2 16x + 32 10. x 2 + 2xy + y 2 11. 36x 2 12x + 1 12. 24z 2 14z 5
SECTION 2: SOLVE QUADRATIC EQUATIONS Solve by factoring. 13. x 2 15x + 56 = 0 14. 15. 16. x 2 + 6x = 0 17. 6x 2 + 11x = 10 18. 19. 20.
Solve by using the square roots method. 21. 2x 2 3 = 43 22. 5(x 4) 2 7 = 53 23. 5x 2 + 9 = 3(2x 2 5) + 1 24. 25. 26.
Solve by completing the square. 27. 3x 2 12x 36= 0 28. x 2 2x 35 = 0 29. 2x 2 = 12x + 46 30. 4x 2 8x = 40 31. 32.
Solve by using the quadratic formula. 33. 2x 2 4x + 3 = 0 34. x 2 + 5x 6 = 0 35. 2x 2 2x + 5 = 2 36. 37. 8 + 5 (x 2 x) = x 2 + 3x 28 38.
Simplify the complex numbers. 39. (3 + 2i) (5 i) 40. (3 + 2i) (5 i) 41. 10 28 42. i 298 43. 4i(5 3i) 44. 1+ i 4 3i 45. a) What is the discriminant? b) If the discriminant is negative, how many solutions will there be and what type? c) If a quadratic equation has 1 real number solution, what will the discriminant be? d) Name as many vocabulary words that are synonymous to an x-intercept of a quadratic function. Give the number and type of solutions each equation has and explain how you know. 46. 47.
SECTION 3: GRAPHING QUADRATIC FUNCTIONS 48. 49. 50. y = 1 2 (x 2)(x + 6) 51.
SECTION 4: SYSTEMS OF EQUATIONS Graph the following systems of equations and inequalities. 54. Solve by graphing. y = x 2 4x + 3 y = x 2 + 12x 27 55. y < x 2 + 4x + 2 y 2x 2 4x 1
57. 58. 59. y = x 2 5x + 10 y = 2x 2 6x + 4 60. x 2 + y 2 16x + 39 = 0 x 2 y 2 9 = 0
SECTION 5: MODELING QUADRATIC FUNCTIONS Write the equation of a quadratic function in vertex form given the vertex and a point. 62. Vertex: ( 4, 5); Point: (0, 27) 63. Vertex: ( 2, 6); Point: (2, 2) Convert to standard form. 64. y = 2 (x + 4) 2 5 65. y = 3 (x + 2) (x 5) Convert to intercept form. 66. y = 18x 2 60x + 50 67. y = 3x 2 + 19x + 14 Convert to vertex form. 68. y = x 2 + 10x + 2 69. y = 3x 2 12x 1
70. The path of a table-tennis ball after being hit and hitting the surface of the table can be modeled by the function h(t ) = 4.9t t 0.42 ( ) where h is the height in meters above the table that the ball is at any time, t. How long does it take the ball to hit the table? What is the maximum height of the ball? When does it reach the maximum height? 71. A concrete pool deck of a uniform width is going to be built around a rectangular pool that is 20 feet long and 15 feet wide. The contractor has enough concrete to over 114 square feet of space. How wide is the deck encompassing the pool? 72. An engineer collects the following data showing the speed, s, of a Ford Taurus and its average miles per gallon, M. Speed, s mpg, m a. What relationship do you notice about the speed of the car and the amount of gas 30 18 consumed? 35 20 40 23 40 25 b. Using you graphing calculator, find the quadratic regression that fits the data. 45 25 50 28 55 30 60 29 c. What speed maximizes your miles per gallon? 65 26 65 25 70 25 d. What is the maximum miles per gallon? e. How many miles per gallon will you get if you drive 63 miles per hour? f. At what speed(s) will you get 20 miles per gallon?
SUMMARY: SOLVING QUADRATIC EQUATIONS 1. Square Roots. Use When: An equation has an x 2 or (x + c) 2 (but does not have an x) 1. Isolate the x 2. 2. Square root both sides. 3. Simplify (including the square root!) 4. Don t forget the ± sign! 2. Factor and Zero Product Property. Use When: The equation is factorable. 1. Make sure the equation is in the form: ax 2 + bx + c = 0 2. Factor completely! 3. Set each factor equal to 0. 4. Solve. 5. Write the solutions together: x =, 3. Complete the Square. 4. Quadratic Formula. Use When: The trinomial is not factorable. A=1 and B is even. Use When: The other methods do not apply. 1. Make sure the equation is in the form: Ax 2 + Bx = C 2. Use the formula B 2 3. Add C to both sides. 2 to determine C. 4. Factor the left side of the equation into a binomial squared. 5. Take the square root of both sides (don t forget ± ) 6. Isolate the x. 1. Put the equation into standard form: Ax 2 + Bx + C = 0 2. Find A, B, C. 3. Substitute A, B, and C into the quadratic formula. Use parentheses! 4. Simplify completely! Quadratic Formula: x = b ± b2 4ac 2a Discriminant : b 2 4ac If negative = 2 imaginary solutions If 0 = one real number solution If positive = 2 real number solutions Recall, i =!1 ZEROS = ROOTS = SOLUTIONS x-intercepts (can only be real numbers) FIND A ZERO (FUNCTION) 1. Substitute 0 for y. Or substitute 0 for f(x). 2. Solve the equation!
Vertex: (h, k) GRAPHING QUADRATIC FUNCTIONS Vertex Form Intercept Form Standard Form ( ) 2 + k y = a x h 1. a > 0: opens up a < 0: opens down Vertex: y = a(x p)(x q) p + q 2, f p + q ( ) 2 1. Find the x-coordinate of the vertex. Vertex: y = ax 2 + bx + c b 2a, f b ( ) 2a 1. Find the x-coordinate of the vertex. 3. Stretch each point (multiply the y-values by a) stretch: a < -1 or a > 1 shrink: -1 < a < 1 4. Use the chart to find other points. 5. Make sure to begin at the key point and that the graph points in the correct direction. 2. Substitute it into the function to find the y-coordinate of the vertex. 3. Use the chart to find other points on the graph. 2. Substitute it into the function to find the y-coordinate of the vertex. 3. Use the chart to find other points on the graph. y = 2 (x + 4) 2 1 y = (x + 4) (x 2) y = x 2 + 4x + 2 Find the x-intercepts (root, solution, zero) Find the y-intercept Substitute 0 for y and solve. Substitute 0 for x and solve.
CONVERTING TO STANDARD FORM Vertex Form: y = a (x h) 2 + k Standard Form: y = ax 2 + bx + c 1. Multiply out the (x h) 2. Remember to FOIL! 2. Distribute the a. 3. Combine any like terms. CONVERTING TO INTERCEPT FORM Standard Form: y = ax 2 + bx + c Intercept Form: y = a (x p) (x q) 1. Factor out a. 2. Factor the trinomial. 3. Taaa daaa! CONVERTING TO VERTEX FORM: COMPLETE THE SQUARE Completing the square is useful for turning a quadratic function in standards form into vertex form. Vertex form is useful for graphing quadratics using transformations! Vertex Form: y = a (x h) 2 + k Standard Form: y = ax 2 + bx + c Method: COMPLETE THE SQUARE! 1. Factor out the a value if necessary. b 2. Use the formula 2 = c 3. Add and subtract that value from the right side. 2 4. Rewrite the trinomial as a binomial squared. 5. Combine any like terms! SYSTEMS OF EQUATIONS Quadratic-Linear: solve the linear for a variable. Substitute into the quadratic. Solve. Quadratic-Quadratic: Use elimination. Solve! REMEMBER!!!! Once you find the x-values, substitute into a function to find the y-values =)
Unit 3 Quadratic Functions: Quick Questions Name: QUESTION ANSWER A ANSWER B 1 State the form of: y = a (x h) 2 + k Vertex Form Standard Form 2 State the form of: y = a (x p) (x q) Intercept Form Standard Form 3 Find the vertex of: y = 3(x + 4) 2 (3, 4) ( 4, 0) 4 5 6 7 Find the x-coordinate of the vertex of: y = 2 (x 6) (x + 2) If the x-coordinate of the vertex is 1, Find the y-coordinate: y = x 2 + 3x Describe the transformation from the parent function of: y = (x 4) + 3 Describe the transformation from the parent function of: y = (x 4) + 3 x = 2 x = 2 y = 4 y = 2 Right 4 Left 4 Up 3 Down 3 8 The graph of y > 2x 2 + 4x + 1 is: Solid Dashed 9 The graph of y > 2x 2 + 4x + 1 is: Shaded Above Shaded Below 10 If the discriminant is 10 there are: 2 irrational solutions 2 imaginary solutions 11 If the discriminant is 0 there are: No Solutions One real solution 12 If the discriminant is 25 there are: 2 rational solutions 2 irrational solutions 13 If the discriminant is 7 there are: 2 rational solutions 2 irrational solutions 14 i 25 1 i 15 16 To complete the square, use the following form: For quadratic formula, use the following form: Ax 2 + Bx + C = 0 Ax 2 + Bx + C = 0 Ax 2 + Bx = C Ax 2 + Bx = C 17 Factor: 9x 2 60x + 100 (3x 10) 2 (3x 10) (3x + 10) 18 19 Find the value of C that would make the trinomial a perfect square: x 2 + 8x + C Find the value of C that would make the trinomial a perfect square: x 2 + 5x + C 16 64 20 The solutions to a system will look like: Ordered pairs: (x, y) x = 25 4 5 2
WHAT S THE NEXT STEP? 1. x 2 +10x + 9 = 0 2. x 2 6x 8 = 0 A. Subtract 9 from both sides. B. Factor and use the zero product property. C. Complete the square. A. Factor into two binomials. B. Divide both sides by 1. C. Add 8 to both sides. 3. 4(x 2 + 5) = x 2 7x 4. 2(x + 4) 2 +1= 7 A. Distribute the 4. B. Divide both sides by 4. C. Factor. A. Multiply out (x + 4) 2. B. Subtract 1 from both sides. C. Divide both sides by 2. 5. 2x 2 +20x + 50 = 0 A. Factor by multiplying 2 times 50. B. Factor out a 2. C. Complete the square. 6. Convert to standard form. y = 2(x + 4) 2 7 A. Distribute the 2. B. Expand the binomial to x 2 + 16. C. Expand the binomial to x 2 + 8x + 16. 7. Convert to intercept form: y = 6x 2 13x 5 A. Complete the Square. B. Factor. 8. Convert to vertex form: y = 2x 2 12x 5 A. Complete the Square. B. Factor. C Multiply. C Multiply.