Quadratic Functions 175

Exercises - Graphs of Quadratic Equations

3
Questions 1 and 2 refer to the following y
information.
y = a ( x − h) 2
y
5
y = x2 − 6 x + 5
4
3
2
1 x
x O (4,0)
−3 −2 −1 O 1 2 3 4 5 6 7
−2
−3 In the xy - plane above, the parabola y = a ( x − h) 2
−4
−5 V has one x- intercept at (4, 0) . If the y - intercept
of the parabola is 9, what is the value of a ?
The graph of quadratic function y = x 2 − 6 x + 5 is
shown above.
-

1
Which of the following is an equivalent form
of the equation of the graph shown above, from
which the coordinates of vertex V can be
identified as constants in the equation? 4
In the xy - plane, if the parabola with equation
A) y = ( x − 1)( x − 5) y = a ( x + 2) 2 − 15 passes through (1,3) , what
B) y = ( x + 1)( x + 5) is the value of a ?

C) y = x( x − 6) + 5

D) y = ( x − 3) 2 − 4

2
Which of the following is an equivalent form 5 -

of the equation of the graph shown above, that

The graph of the equation y = a ( x − 1)( x + 5) is
displays the x- intercepts of the parabola as
constants? a parabola with vertex (h, k ) . If the minimum
value of y is −12 , what is the value of a ?
A) y = ( x − 1)( x − 5)

B) y = ( x + 1)( x + 5)

C) y = x( x − 6) + 5

D) y = ( x − 3) 2 − 4
178 Chapter 11

Exercises – Factoring Trinomials

1 4

x 2 − 2 x − 24 3x 2 + 7 x − 6

Which of the following is equivalent to the Which of the following is equivalent to the
expression above? expression above?

A) ( x + 3)( x − 8) A) (3 x − 2)( x + 3)

B) ( x − 3)( x + 8) B) (3 x + 2)( x − 3)

C) ( x − 6)( x + 4) C) (3 x − 1)( x + 6)

D) ( x + 6)( x − 4) D) (3 x + 1)( x − 6)

2 5

x 2 − 17 x + 72 2 x 2 + x − 15

Which of the following is equivalent to the Which of the following is equivalent to the
expression above? expression above?

A) ( x + 8)( x − 9) A) (2 x + 3)( x − 5)

B) ( x − 8)( x − 9) B) (2 x − 3)( x + 5)

C) ( x − 12)( x − 6) C) (2 x − 5)( x + 3)

D) ( x − 12)( x + 6) D) (2 x + 5)( x − 3)

3 6

− x 2 + 5 x + 84 −6 x 2 + x + 2

Which of the following is equivalent to the Which of the following is equivalent to the
expression above? expression above?

A) (12 − x)( x + 7) A) −(6 x + 1)( x − 2)

B) (12 + x)( x − 7) B) −(6 x − 1)( x + 2)

C) (21 + x)( x − 4) C) −(3 x + 2)(2 x − 1)

D) (21 − x)( x + 4) D) −(3 x − 2)(2 x + 1)

180 Chapter 11

Exercises – Factoring Differences of Squares and Perfect Square Trinomials

1 4

3 x 2 − 48 If x + y = 10 and x − y = 4 , what is the value

Which of the following is equivalent to the of x 2 − y 2 ?

expression above?
A) 20
A) 3( x − 4)( x + 4) B) 24
C) 36
B) 3( x − 4) 2
D) 40
C) (3 x − 4)( x + 4)

D) (3 x + 4)( x − 4)
5

6 x 2 + 7 x − 24 = 0
2 If r and s are two solutions of the equation
above and r > s , which of the following is the
x − 6 x − 16
value of r − s ?
Which of the following is equivalent to the
expression above? 7
A)
6

A) ( x − 4) 2 16
B)
3
B) ( x − 4)( x + 4)
25
C)
C) ( x + 8)( x − 2) 6

D) ( x − 8)( x + 2) 20
D)
3

3 6

If x 2 + y 2 = 10 and xy = −3 , what is the value x 2 − 3 x = 28

of ( x − y ) 2 ?
If r and s are two solutions of the equation
above, which of the following is the value of
A) 12 r+s?
B) 16
A) −3
C) 20
B) 3
D) 25
C) 6
D) 9
182 Chapter 11

Exercises - Solving Quadratic Equations by Completing the Square

1 4

If x 2 − 10 x = 75 and x < 0 , what is the value k2

x 2 − rx =
of x + 5 ? 4

A) −15 In the quadratic equation above, k and r are

are constants. What are the solutions for x ?
B) −10
C) −5
r k 2 + 2r 2
D) 0 A) x = ±
4 4

r k 2 + 8r 2
B) x = ±
2 4
2
r k2 + r2
k C) x = ±
If x 2 − kx = 20 and x − = 6 , which of the 4 2
2
following is a possible value of x ? r k2 + r2
D) x = ±
2 2
A) 2
B) 4
C) 6 5
D) 8
If ( x − 7)( x − s ) = x 2 − rx + 14 for all values of x ,
what is the value of r + s ?

3
k
x2 − x=5
3

Which of the following is an equivalent form of the

equation shown above, from which the equation
could be solved by completing the square?
6
k k k 3
A) x 2 − x + = + 5 If x 2 − x + c = ( x − k ) 2 , what is the value of c ?
3 6 6 2
k k2 k2
B) x 2 − x+ = +5
3 9 9

k k2 k2
C) x 2 − x+ = +5
3 36 36

k k2 k2
D) x 2 − x+ = +5
3 6 6
Quadratic Functions 185

Exercises - Quadratic Formula and the Discriminant

1 4

( p − 1) x 2 − 2 x − ( p + 1) = 0  y = bx − 3
 2
 y = ax − 7 x
In the quadratic equation above, p is a constant.
What are the solutions for x ? In the system of equations above, a and b are
constants. For which of the following values of
a and b does the system of equations have exactly
1 + 2 − p2 1 − 2 − p2 two real solutions?
A) and
p −1 p −1

1+ 2 p A) a = 3, b = −2
B) and −1
p −1 B) a = 5, b = 0
p +1 C) a = 7, b = 2
C) and −1
p −1
D) a = 9, b = 4
p +1 2 p +1
D) and
p −1 p −1

2 5
What is the sum of all values of x that satisfy What are the solutions to x 2 + 4 = −6 x ?
3 x 2 + 12 x − 29 = 0 ?
A) −3 ± 13
A) −4
B) −3 ± 5
B) −2
C) 2 C) −6 ± 5

D) 4 D) −6 ± 13

3 6

If the quadratic equation kx 2 + 6 x + 4 = 0 has Which of the following equations has no real
exactly one solution, what is the value of k ? solution?

3 A) 5 x 2 − 10 x = 6
A)
2 B) 4 x 2 + 8 x + 4 = 0
5
B) C) 3 x 2 − 5 x = −3
2
1
7 D) − x 2 + 2 x − 2 = 0
C) 3
4
9
D)
4
Quadratic Functions 187

Exercises - Solving Systems Consisting Linear and Quadratic Equations

1 3
y y = f ( x)
V
16
(0, )
3
( − a, b) ( a, b)

y = g ( x)

x x
(−4,0) O (2,0) O

The function f and g , defined by f ( x) = 2 x 2 + 2

The xy - plane above shows two x- intercepts, and g ( x) = −2 x 2 + 18 , are graphed in the xy - plane
a y - intercept and vertex V of a parabola. If above. The two graphs intersect at the points (a, b)
the line passes through the points (2, 0) and V , and (−a, b) . What is the value of b ?
which of the following must be the y - intercept
of the line?
A) 6

A) 3 B) 8

7 C) 10
B)
2 D) 12
C) 4
9
D)
2
4

 x + y = 14
2 2

 2
2  x − y = 2

 y = x + x
2
 If ( x, y ) is a solution to the system of equations
 y = ax − 1
above, what is the value of x 2 ?

In the system of equations above, a > 0 . If the

system of equations has exactly one real solution, A) 2
what is the value of a ? B) 3
C) 4
5
A) D) 5
2
B) 3
7
C)
2
D) 4
188 Chapter 11

Chapter 11 Practice Test

1 3
y y

B C
(0,6)

x
(−2,0) O (6,0) x
A(−1,0) O (3,0) D(8,0)

The graph of the quadratic function above shows

In the figure above, the vertex of the graph of
two x- intercepts and a y - intercept. Which of
the quadratic function is at (3, 0) . The points
the following equations represents the graph of
the quadratic function above? B and C lie on the parabola. If ABCD is a
rectangle with perimeter 38, which of the
following represents the equation of the parabola?
1
A) y = − ( x − 1) 2 + 9
2
2
A) y = ( x − 3) 2
1 5
B) y = − ( x − 2) 2 + 8
2
5
B) y = ( x − 3) 2
1 8
C) y = − ( x − 2) 2 + 9
2
3
C) y = ( x − 3) 2
1 4
D) y = − ( x − 3) 2 + 8
2
7
D) y = ( x − 3) 2
8

If ( x + y ) 2 = 324 and ( x − y ) 2 = 16 , what is the 4

value of xy ?
If (ax + b)(2 x − 5) = 12 x 2 + kx − 10 for all values
of x , what is the value of k ?
A) 33
B) 55 A) −26

C) 77 B) −10

D) 99 C) 24
D) 32
Quadratic Functions 189

7
Questions 5-8 refer to the following
information. What is the maximum height from the ground the
object will reach, to the nearest meter?
1
h = − gt 2 + v0 t + h0
2 A) 103
B) 112
The equation above describes the motion of an object
thrown upward into the air. In the equation, g is the C) 125
acceleration due to gravity (9.8m/s 2 ) , t is the time D) 133
elapsed since the object was thrown upward, v0 is
the initial speed of the object, h0 is the initial height
from which the object was thrown, and h is the height
of the object above the ground t seconds after the
8
object was thrown.
How long will it take the object to hit the ground,
to the nearest second? (hint: Height of the object
is zero when the object hits the ground.)
5
Which of the following equations represents the A) 7
motion of the object, if the object was thrown
B) 8
upward from 40 meters above the ground with
an initial speed of 35 meters per second (m/s) ? C) 9
D) 10
2
A) h = −9.8t + 40t + 35

B) h = −9.8t 2 + 35t + 40

C) h = −4.9t 2 + 40t + 35

D) h = −4.9t 2 + 35t + 40
9

h = −16t 2 + h0

6 The equation above describes the height of an

object t seconds after it dropped from a height
How many seconds will it take the object to reach
of h0 feet above the ground. If a hiker dropped
its maximum height? (hint: The function has a
b a water bottle from a cliff 150 feet above the
maximum point at t = − .) ground, how many seconds will it take to hit the
2a ground? (Round your answer to the nearest
second.)
15
A)
7
A) 2
20 B) 3
B)
7
C) 4
25 D) 5
C)
7
30
D)
7