QUADRATIC FUNCTIONS TEST REVIEW NAME: __________________________

SECTION 1: FACTORING Factor each expression completely. 1. 3x 2 − 48 2. 25p2 – 16p

3. 6x 2 −13x − 5 4. 9x 2 − 30x + 25

5. 4x2 + 81 6. 6x2 – 14x + 4

7. 4x2 + 20x – 24 8. 4x2 + 20x + 25

9. 2x 2 −16x + 32 10. x2 + 2xy + y2

11. 36x2 – 12x + 1 12. 24z2 – 14z – 5

SECTION 2: SOLVE QUADRATIC EQUATIONS Solve by factoring. 13. x2 −15x +56 = 0 14.

15. 16. x2 + 6x = 0

17. 6x2 + 11x = 10 18.

19.

20.

Solve by using the square roots method. 21. 2x2 – 3 = –43 22. 5(x – 4)2 – 7 = 53

23. 5x2 + 9 = 3(2x2 – 5) + 1

24.

25.

26.

Solve by completing the square. 27. 3x2 – 12x – 36= 0 28. x2 – 2x – 35 = 0

29. 2x2 = –12x + 46 30. 4x2 – 8x = 40

31. 32.

Solve by using the quadratic formula. 33. 2x2 – 4x + 3 = 0

34. x2 + 5x – 6 = 0

35. 2x2 – 2x + 5 = 2

36.

37. 8 + 5 (x2 – x) = x2 + 3x –28 38.

Simplify the complex numbers.

39. (3 + 2i) – (5 – i) 40. (3 + 2i) (5 – i)

41. 10− −28 42. i298

43. 4i(5 – 3i)

44.

1+ i4−3i

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 = x2 – 4x + 3 y = –x2 + 12x – 27

55. y < −x2 + 4x+ 2 y ≥ 2x2 − 4x−1

57.

58.

59. y = x2 – 5x + 10 y = 2x2 – 6x + 4

60 . x2 + y2 – 16x + 39 = 0 x2 – y2 – 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 = 18x2 – 60x + 50 67. y = –3x2 + 19x + 14

Convert to vertex form. 68. y = x2 + 10x + 2 69. y = 3x2 – 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.

a. What relationship do you notice about the speed of the car and the amount of gas consumed?

b. Using you graphing calculator, find the quadratic regression that fits the data.

c. What speed maximizes your miles per gallon?

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?

Speed, s mpg, m 30 18 35 20 40 23 40 25 45 25 50 28 55 30 60 29 65 26 65 25 70 25

SUMMARY: SOLVING QUADRATIC EQUATIONS

1. Square Roots. Use When: An equation has an x2 or (x + c)2 (but does not have an x) 1. Isolate the x2. 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: ax2 + 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. Use When: The trinomial is not factorable. A=1 and B is even. 1. Make sure the equation is in the form: Ax2 + Bx = C

2. Use the formula

B2

⎛

⎝⎜⎞

⎠⎟

2

to determine C.

3. Add C to both sides. 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.

4. Quadratic Formula. Use When: The other methods do not apply. 1. Put the equation into standard form: Ax2 + 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 : b2 – 4ac If negative = 2 imaginary solutions If 0 = one real number solution If positive = 2 real number solutions

Recall, i =

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!

±

±

!1

GRAPHING QUADRATIC FUNCTIONS

Vertex Form Intercept Form Standard Form

y = a x − h( )2 +k

Vertex: (h, k) 1. a > 0: opens up a < 0: opens down 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.

y = a(x − p)(x − q )

Vertex:

p + q

2, f

p + q

2( ) ⎛

⎝⎜⎞⎠⎟

1. Find the x-coordinate of the vertex. 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 = ax 2 + bx + c

Vertex:

−b2a

, f−b

2a( ) ⎛

⎝⎜⎞⎠⎟

1. Find the x-coordinate of the vertex. 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 = x2 + 4x + 2

Find the x-intercepts (root, solution, zero) Substitute 0 for y and solve.

Find the y-intercept Substitute 0 for x and solve.

CONVERTING TO STANDARD FORM

Vertex Form: y = a (x – h)2 + k Standard Form: y = ax2 + 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 = ax2 + 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 = ax2 + bx + c

Method: COMPLETE THE SQUARE! 1. Factor out the a value if necessary.

2. Use the formula

b2

⎛⎝⎜

⎞⎠⎟

2

= c

3. Add and subtract that value from the right side. 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 Find the x-coordinate of the vertex of: y = 2 (x – 6) (x + 2)

x = 2 x = –2

5 If the x-coordinate of the vertex is –1, Find the y-coordinate: y = –x2 + 3x

y = –4 y = –2

6 Describe the transformation from the parent function of: y = (x – 4) + 3

Right 4 Left 4

7 Describe the transformation from the parent function of: y = (x – 4) + 3

Up 3 Down 3

8 The graph of y > 2x2 + 4x + 1 is: Solid Dashed

9 The graph of y > 2x2 + 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 i25 –1 i

15 To complete the square, use the following form:

Ax2 + Bx + C = 0 Ax2 + Bx = C

16 For quadratic formula, use the following form:

Ax2 + Bx + C = 0 Ax2 + Bx = C

17 Factor: 9x2 – 60x + 100 (3x – 10)2 (3x – 10) (3x + 10)

18 Find the value of C that would make the trinomial a perfect square: x2 + 8x + C

16 64

19 Find the value of C that would make the trinomial a perfect square: x2 + 5x + C

254

52

20 The solutions to a system will look like: Ordered pairs: (x, y) x = _____

WHAT’S THE NEXT STEP? 1. x2 +10x + 9 = 0

A. Subtract 9 from both sides. B. Factor and use the zero product property. C. Complete the square.

2. −x 2 −6x −8= 0

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 A. Distribute the 4. B. Divide both sides by 4. C. Factor.

4. 2(x + 4)2 +1= 7

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 x2 + 16. C. Expand the binomial to x2 + 8x + 16.

7. Convert to intercept form:

y = 6x 2 −13x − 5

A. Complete the Square. B. Factor. C Multiply.

8. Convert to vertex form:

y = 2x 2 −12x − 5 A. Complete the Square. B. Factor. C Multiply.