Identify the vertex and axis of symmetry of each. Then sketch the graph.
15) f (x) = −3( x − 2) 2 − 4
16) f (x) = −
y
1
( x − 1) 2 + 4
4
8
y
6
8
4
6
2
4
2
−8
−6
−4
−2
2
4
6
8 x
−2
−8
−6
−4
−2
2
−4
−2
−6
−4
−8
−6
4
6
8 x
4
6
8 x
4
6
8 x
−8
17) f (x) =
1
( x + 4) 2 + 3
4
18) f (x) =
1
( x + 5) 2 + 2
4
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
19) f (x) = −2( x + 5) 2 − 3
20) f (x) = ( x + 2) 2 − 1
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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-2-
Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 2
Name___________________________________
Properties of Parabolas
Date________________ Period____
Identify the vertex of each.
1) y = x 2 + 16 x + 64
2) y = 2 x 2 − 4 x − 2
3) y = − x 2 + 18 x − 75
4) y = −3 x 2 + 12 x − 10
Graph each equation.
5) y = x 2 − 2 x − 3
6) y = − x 2 − 6 x − 10
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
6
8 x
Identify the min/max value of each. Then sketch the graph.
7) f (x) = − x 2 + 8 x − 20
1
4
16
8) f (x) = − x 2 + x −
3
3
3
y
8
y
−8
−6
−4
6
8
4
6
2
4
−2
2
4
6
2
8 x
−2
−8
−6
−4
−2
2
−4
−2
−6
−4
−8
−6
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4
−8
-1-
Worksheet by Kuta Software LLC
9) f (x) = x 2 + 2 x − 1
10) f (x) = − x 2 − 10 x − 30
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
Identify the vertex, axis of symmetry, and min/max value of each.
11) f (x) = 3 x 2 − 54 x + 241
12) f (x) = x 2 − 18 x + 86
4
48
114
13) f (x) = − x 2 +
x−
5
5
5
14) f (x) = −2 x 2 − 20 x − 46
1
15) f (x) = − x 2 + 7
4
16) f (x) = x 2 − 12 x + 44
1
17) f (x) = x 2 − x + 9
4
18) f (x) = x 2 + 4 x + 5
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-2-
Worksheet by Kuta Software LLC
Name Class Date Practice
Form K
Factoring Quadratic Expressions
Factor each expression.
1. x2 + 4x - 5
2. x2 + 13x + 42
3. -x2 - x + 12
4. x2 - 8x + 16
5. -x2 + 16x - 55
6. x2 + 2x - 48
7. -y 2 + 17y - 72
8. x2 + 7x + 12
9. x2 - 8x + 12
Find the GCF of each expression. Then factor the expression.
10. 3x 2 + 15x + 12
11. -9y 2 + 6y
12. 6x 2 + 12x - 48
13. -3x2 - 3x + 60
14. 2x 2 - 10x
15. 7x 2 - 14x - 56
16. 10x 2 + 100x
17. 9x 2 - 36x + 27
18. -5xy 2 - 30xy - 25x
19. Writing When you factor a quadratic expression, explain what it means when
c 6 0 and b 7 0.
20. Error Analysis You factored -x2 + 10x - 24 as -(x - 6)(x - 4). Your friend factored
it as (x + 12)(x - 2). Which of you is correct? What mistake was made?
21. Multiple Choice What is the factored form of -14a2 + 42ab?
a(-14a + 42b)
7(-2a2 + 6ab)
-2a(7a - 21b)
-14a(a - 3b)
22. Reasoning The area of a carpet is (x2 - 11x + 28) ft2 . What are the length and
the width of the carpet?
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice (continued)
Form K
Factoring Quadratic Expressions
Factor each expression.
23. 2x 2 + 7x + 6
24. 3x 2 - 14x - 24
25. 5x 2 - 22x + 21
26. 4x 2 + 18x + 8
27. 2x 2 - 8x + 6
28. 6x 2 + 13x - 28
29. 4x 2 - 4x + 1
30. x2 + 6x + 9
31. 4x 2 - 16
32. 9x 2 - 4
33. 16x 2 - 40x + 25
34. x2 - 25
35. 9x 2 - 36x + 36
36. 25x 2 - 9
37. 4x 2 + 24x + 36
38. Error Analysis Which of the following examples is factored correctly? Explain.
Example 1
Example 2
4x2
4x2 - 49
(2x)2 - 72
(2x - 7)(2x - 7)
- 49
2
(2x) - 72
(2x - 7)(2x + 7)
39. You can represent the area of a square tabletop with the expression
16x2 + 24x + 9. What is the side length of the tabletop in terms of x?
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice
Form K
Quadratic Equations
Solve each equation by factoring. Check your answers. To start, factor the
quadratic expression.
1. x2 - x - 30 = 0
2. x2 - 10x = -21
3. x2 = -10x - 9
4. x2 - 5x = 0
5. 10x - 24 = x2
6. x2 = -12x
Solve each equation using tables. Give each answer to at most two decimal
places. To start, enter the equation as Y1. Make a table and look for where the
y-values change sign.
7. x2 + x = 12
8. 10x 2 + 26x + 16 = 0
10. 2x 2 - 13x + 18 = 0
11. 2x 2 = 10x
9. 2x 2 + 11x = 6
12. 0.5x 2 - 8 = 0
Write a quadratic equation with the given solutions.
13. 4 and -5
14. -6 and 0
15. 3 and 8
16. Writing Explain when you would prefer to use factoring to solve a quadratic
equation and when you would prefer to use tables.
17. A parabolic jogging path intersects both ends of a street. The path has the
equation x2 - 25x = 0. If one end of the street is considered to be x = 0 and
the street lies on the x-axis, where else does the path intersect the street?
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice (continued)
Form K
Quadratic Equations
Solve each equation by graphing. Give each answer to at most two decimal
places.
18. 2x 2 - x - 10 = 0
19. 6x 2 - 13x = 28
20. 4x 2 + 27x = 12
21. 4x 2 - 5x - 26 = 0
22. 6x 2 - 23x = 18
23. 4x 2 - 9x + 5 = 0
24. The students in Mr. Wilson’s Physics class are making golf ball catapults. The
flight of group A’s ball is modeled by the equation y = -0.014x2 + 0.68x,
where x is the ball’s distance from the catapult. The units are in feet.
a. How far did the ball fly?
b. How high above the ground did the ball fly?
c. What is a reasonable domain and range for this function?
25. A rectangular pool is 20 ft wide and 50 ft long. The pool is surrounded by a
walkway. The walkway is the same width all the way around the pool. The total
area of the walkway is 456 square ft. How wide is the walkway?
26. Reasoning The equation used to solve Exercise 25 has two solutions. Why is
only one solution used to answer the question?
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice
Form K
Completing the Square
Solve each equation by finding square roots. To start, remember to isolate x2.
1. x2 - 9 = 0
2. x2 + 4 = 20
3. x2 + 15 = 16
4. 2x 2 - 64 = 0
5. 4x 2 - 100 = 0
6. 5x 2 - 25 = 0
x2
x2
=9
= 16
x2 = 1
7. You are painting a large wall mural. The wall length is 3 times the height. The
area of the wall is 300 ft2.
a. What are the dimensions of the wall?
b. If each can of paint covers 22 ft2, will 12 cans be enough to cover the wall?
8. The lengths of the sides of a carpet have the ratio of 4.4 to 1. The area of the
carpet is 1154.7 ft2. What are the dimensions of the carpet?
9. A packing box is 4 ft deep. One side of the box is 1.5 times longer than the
other. The volume of the box is 24 ft3. What are the dimensions of the box?
Solve each equation. To start, factor the perfect square trinomial.
10. x2 - 14x + 49 = 81
(x - 7)2 = 81
13. 4x 2 + 36x + 81 = 16
11. x2 + 6x + 9 = 1
12. 9x 2 - 12x + 4 = 49
(x + 3)2 = 1
(3x - 2)2 = 49
14. x2 + 2x + 1 = 36
15. x2 - 16x + 64 = 9
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice (continued)
Form K
Completing the Square
Complete the following squares.
16. x2 + 8x +
e
e
( 82 )2 = 42 = 16
19. x2 - 24x +
e
17. x2 + 20x +
( 202 )2 =
e
20. x2 + 34x +
e
e
18. x2 - 14x +
e
e
e
e
21. x2 - 46x +
e
e
Solve the following equations by completing the square.
22. x2 - 8x - 5 = 0
23. x2 + 12x + 9 = 0
x2 - 8x = 5
x2 - 8x + 16 = 5 + 16
(x - 4)2 = 21
x - 4 = { 121
x=
e
24. x2 - 10x = -11
x2 + 12x = -9
x2 + 12x + 36 = -9 + 36
e
25. 2x 2 + 11x - 23 = -x + 3
26. x2 - 18x + 64 = 0
27. 3x 2 - 42x + 78 = 0
Write the following equations in vertex form.
28. y = x2 + 10x - 9
29. y = x2 - 18x + 13
30. y = x2 + 32x - 8
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice
Form K
The Quadratic Formula
Solve each equation using the Quadratic Formula. To start, find the values of a, b,
and c. Substitute those values into the Quadratic Formula. When necessary round
real solutions to the nearest hundredth.
1. x2 - 4x + 3 = 0
a = 1, b = -4, c = 3
2. 2x 2 + 3x - 4 = 0
a = 2, b = 3, c = -4
- ( - 4) { 2( - 4)2 - (4)(1)(3)
2(1)
4. x2 + 3x = 3
- (3) { 2(3)2 - (4)(2)( - 4)
2(2)
5. 4x 2 + 3 = 9x
3. 8x 2 - 2x - 5 = 0
a = 8, b = -2, c = -5
- ( - 2) { 2( - 2)2 - (4)(8)( - 5)
2(8)
6. 2x - 5 = -x2
7. Your school sells yearbooks every spring. The total profit p made depends on
the amount x the school charges for each yearbook. The profit is modeled by the
equation p = -2x2 + 70x + 520. What is the smallest amount in dollars the school
can charge for a yearbook and make a profit of at least $1000?
To start, substitute 1000 for p in the equation. 1000 = -2x2 + 70x + 520
Then, write the equation in standard form. 2x2 - 70x + 480 = 0
8. Engineers can use the formula d = 0.05s2 + 1.1s to estimate the minimum stopping
distance d in feet for a vehicle traveling s miles per hour.
a. If a car can stop after 65 feet, what is the fastest it could have been traveling
when the driver put on the brakes?
b. Reasoning Explain how you knew which of the two solutions from the
Quadratic Formula to use. (Hint: Remember this is a real situation.)
9. Reasoning Explain why a quadratic equation has no real solutions if the
discriminant is less than zero.
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice (continued)
Form K
The Quadratic Formula
Evaluate the discriminant for each equation. Determine the number of real
solutions.
10. -12x 2 + 5x + 2 = 0
11. x2 - x + 6 = 0
12. 2x - 5 = -x2
13. 4x 2 + 7 = 9x
14. x2 - 4x = -4
15. 3x + 6 = -6x2
(5)2
- 4(-12)(2)
(-1)2
- 4(1)(6)
(2)2 - 4(1)(-5)
Solve each equation using any method. When necessary, round real solutions
to the nearest hundredth.
16. 7x 2 + 3x = 12
17. x2 + 6x - 7 = 0
7x2 + 3x - 12 = 0
(x + 7)(x - 1) = 0
- 3 { 2(3)2 - 4(7)( - 12)
2(7)
19. -12x + 7 = 5 - 2x2
18. 5x = -3x2 + 2
-3x2 - 5x + 2 = 0
5 { 2(5)2 - 4( - 3)(2)
2( - 3)
20. 9x 2 - 6x - 4 = -5
21. 2x - 24 = -x2
Without graphing, determine how many x-intercepts each function has.
22. y = 2x 2 - 3x + 5
23. y = 2x 2 - 4x + 1
24. y = x2 + 3x + 3
25. y = 9x 2 - 12x + 7
26. y = -5x 2 + 8x - 3
27. y = x2 + 16x + 64
(3)2 - 4(2)(5)
(-4)2 - 4(2)(1)
(3)2 - 4(1)(3)
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.