Section 5.2
Vertex Form
453
5.2 Exercises
In Exercises 1-8, expand the binomial.
4
x+
5
2
2
2.
4
x−
5
3.
1.
16.
x2 + 10x + 25
In Exercises 17-24, transform the given
quadratic function into vertex form f (x) =
(x − h)2 + k by completing the square.
17.
f (x) = x2 − x + 8
(x + 3)2
18.
f (x) = x2 + x − 7
4.
(x + 5)2
19.
f (x) = x2 − 5x − 4
5.
(x − 7)2
20.
f (x) = x2 + 7x − 1
21.
f (x) = x2 + 2x − 6
22.
f (x) = x2 + 4x + 8
23.
f (x) = x2 − 9x + 3
24.
f (x) = x2 − 7x + 8
6.
7.
2
(x − 6)2
8.
2
x−
5
5
x−
2
2
In Exercises 25-32, transform the given
In Exercises 9-16, factor the perfect square quadratic function into vertex form f (x) =
trinomial.
a(x − h)2 + k by completing the square.
9.
10.
1
6
9
x2 − x +
5
25
25
x + 5x +
4
2
25.
f (x) = −2x2 − 9x − 3
26.
f (x) = −4x2 − 6x + 1
27.
f (x) = 5x2 + 5x + 5
28.
f (x) = 3x2 − 4x − 6
29.
f (x) = 5x2 + 7x − 3
11.
x2 − 12x + 36
12.
x2 + 3x +
13.
x2 + 12x + 36
30.
f (x) = 5x2 + 6x + 4
14.
3
9
x2 − x +
2
16
31.
f (x) = −x2 − x + 4
x2 + 18x + 81
32.
f (x) = −3x2 − 6x + 4
15.
9
4
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
Version: Fall 2007
454
Chapter 5
Quadratic Functions
In Exercises 33-38, find the vertex of
the graph of the given quadratic function.
33.
f (x) = −2x2 + 5x + 3
34.
f (x) = x2 + 5x + 8
35.
f (x) = −4x2 − 4x + 1
36.
f (x) = 5x2 + 7x + 8
37.
f (x) = 4x2 + 2x + 8
38.
f (x) = x2 + x − 7
In Exercises 39-44, find the axis of symmetry of the graph of the given quadratic
function.
iv. Set up a table near your coordinate
system that calculates the coordinates
of two points on either side of the axis
of symmetry. Plot these points and
their mirror images across the axis of
symmetry. Draw the parabola and
label it with its equation
v. Use the graph of the parabola to determine the domain and range of the
quadratic function. Describe the domain and range using interval notation.
45.
f (x) = x2 − 8x + 12
46.
f (x) = x2 + 4x − 1
47.
f (x) = x2 + 6x + 3
48.
f (x) = x2 − 4x + 1
49.
f (x) = x2 − 2x − 6
50.
f (x) = x2 + 10x + 23
39.
f (x) = −5x2 − 7x − 8
40.
f (x) = x2 + 6x + 3
41.
f (x) = −2x2 − 5x − 8
51.
f (x) = −x2 + 6x − 4
42.
f (x) = −x2 − 6x + 2
52.
f (x) = −x2 − 6x − 3
43.
f (x) = −5x2 + x + 6
53.
f (x) = −x2 − 10x − 21
44.
f (x) = x2 − 9x − 6
54.
f (x) = −x2 + 12x − 33
For each of the quadratic functions in
Exercises 45-66, perform each of the
following tasks.
55.
f (x) = 2x2 − 8x + 3
56.
f (x) = 2x2 + 8x + 4
i. Use the technique of completing the
square to place the given quadratic
function in vertex form.
ii. Set up a coordinate system on a sheet
of graph paper. Label and scale each
axis.
iii. Draw the axis of symmetry and label
it with its equation. Plot the vertex
and label it with its coordinates.
57.
f (x) = −2x2 − 12x − 13
58.
f (x) = −2x2 + 24x − 70
59.
f (x) = (1/2)x2 − 4x + 5
60.
f (x) = (1/2)x2 + 4x + 6
61.
f (x) = (−1/2)x2 − 3x + 1/2
62.
f (x) = (−1/2)x2 + 4x − 2
Version: Fall 2007
Section 5.2
63.
f (x) = 2x2 + 7x − 2
64.
f (x) = −2x2 − 5x − 4
−3x2
+ 8x − 3
65.
f (x) =
66.
f (x) = 3x2 + 4x − 6
Vertex Form
455
79. Evaluate f (4x − 1) if f (x) = 4x2 +
3x − 3.
80. Evaluate f (−5x−3) if f (x) = −4x2 +
x + 4.
In Exercises 67-72, find the range of
the given quadratic function. Express
your answer in both interval and set notation.
67.
f (x) = −2x2 + 4x + 3
68.
f (x) = x2 + 4x + 8
69.
f (x) = 5x2 + 4x + 4
70.
f (x) = 3x2 − 8x + 3
71.
f (x) = −x2 − 2x − 7
72.
f (x) = x2 + x + 9
Drill for Skill. In Exercises 73-76,
evaluate the function at the given value
b.
73.
f (x) = 9x2 − 9x + 4; b = −6
74.
f (x) = −12x2 + 5x + 2; b = −3
75.
f (x) = 4x2 − 6x − 4; b = 11
76.
f (x) = −2x2 − 11x − 10; b = −12
Drill for Skill. In Exercises 77-80,
evaluate the function at the given expression.
77. Evaluate f (x + 4) if f (x) = −5x2 +
4x + 2.
78. Evaluate f (−4x−5) if f (x) = 4x2 +
x + 1.
Version: Fall 2007
456
Chapter 5
Quadratic Functions
5.2 Answers
1.
8
16
x + x+
5
25
3.
x2 + 6x + 9
5.
x2 − 14x + 49
7.
x2 − 12x + 36
3 2
x−
5
2
11.
(x − 6)2
13.
(x + 6)2
15.
(x + 9)2
1
x−
2
2
5
x−
2
2
17.
19.
21.
−
9
x−
2
5 x+
−
33.
x=−
7
10
41.
x=−
5
4
43.
x=
45.
f (x) = (x − 4)2 − 4
1
10
y
f (x)=x2 −8x+12
41
4
1
2
2
+
+
−
2
Version: Fall 2007
57
8
15
4
2
1
−1 x +
2
5 49
,
4 8
(4,−4)
69
4
2
7
5 x+
10
31.
39.
x
10
2
9
−2 x +
4
29.
(x + 1) − 7
27.
1 31
− ,
4 4
10
31
+
4
25.
2
23.
1
− ,2
2
35.
37.
9.
+
109
20
17
4
x=4
Domain = R, Range = [−4, ∞)
Section 5.2
47.
f (x) = (x + 3)2 − 6
f (x)=x2 +6x+3
10
51.
Vertex Form
457
f (x) = −(x − 3)2 + 5
y
10
y
x=3
(3,5)
x
10
x
10
(−3,−6)
f (x)=−x2 +6x−4
x=−3
Domain = R, Range = (−∞, 5]
Domain = R, Range = [−6, ∞)
49.
f (x) = (x − 1)2 − 7
10
53.
y f (x)=x2 −2x−6
f (x) = −(x + 5)2 + 4
x=−5
10
y
(−5,4)
x
10
x
10
(1,−7)
x=1
Domain = R, Range = [−7, ∞)
f (x)=−x2 −10x−21
Domain = R, Range = (−∞, 4]
Version: Fall 2007
458
55.
Chapter 5
Quadratic Functions
f (x) = 2(x − 2)2 − 5
10
y
59.
f (x) = (1/2)(x − 4)2 − 3
f (x)=2x2 −8x+3
10
y
f (x)=(1/2)x2 −4x+5
x
10
x
10
(4,−3)
(2,−5)
x=2
x=4
Domain = R, Range = [−5, ∞)
57.
Domain = R, Range = [−3, ∞)
f (x) = −2(x + 3)2 + 5
x=−3
10
61.
f (x) = (−1/2)(x + 3)2 + 5
y
10
(−3,5)
(−3,5)
x
10
f (x)=−2x2 −12x−13
Domain = R, Range = (−∞, 5]
Version: Fall 2007
y
x
10
x=−3
f (x)=(−1/2)x2 −3x+1/2
Domain = R, Range = (−∞, 5])
Section 5.2
63.
Vertex Form
459
f (x) = 2(x + 7/4)2 − 65/8
10
y
f (x)=2x2 +7x−2
x=−7/4
x
10
(−7/4,−65/8)
Domain = R, Range = [−65/8, ∞)
65.
f (x) = −3(x − 4/3)2 + 7/3
10
y
(4/3,7/3)
x
10
x=4/3
f (x)=−3x2 +8x−3
Domain = R, Range = (−∞, 7/3]
67.
69.
(−∞, 5] = {x |x ≤ 5 }
16
16
, ∞ = x x ≥
5
5
71.
(−∞, −6] = {x |x ≤ −6 }
73.
382
75.
414
77.
−5x2 − 36x − 62
79.
64x2 − 20x − 2
Version: Fall 2007