Math 20-1 – Pre-calculus 11
Quadratic Functions Review
Multiple Choice Questions
Choose the single best response to each of the following. Place your answer under PART A of the answer
sheet provided.
1.
The quadratic function y   x 2  m would have its vertex at;
(1, m)
A.
(0, m)
B.
(1,m)
C.
(0,m)
D.
2.
A quadratic function has a range such that y  4, y  R , with a domain x  R . The number of xintercepts of this function must be;
A.
0
B.
1
C.
2
D.
3
10
3.
The graph shown to the right is of the function
f ( x)  ( x  m) 2  n . The function g ( x)  ( x  m) 2  n is best
described as;
A.
Sharing the same vertex and range as f (x) .
B.
Sharing the same vertex, but having a range y  n, y  R
C.
Having a different vertex then f (x) , but sharing the
same range as f (x) .
D.
Having a different vertex then f (x) , and having a range
y  n, y  R .
f x
2
4
x
4
4.
A rectangle has a width of 2  x , and a length of 4x . The maximum area of this rectangle must be;
A.
4 units2
B.
6 units2
C.
8 units2
D.
16 units2
5.
A quadratic function with y-intercept at (0,20) has a minimum at 2 when x  3 . When written in the
form y  a( x  p) 2  q , the value of a must be;
A.
–2
B.
2
1
C.
2
1
D.
2
6.
7.
5
Three functions, all of the form y  ax 2 , are shown to the right.
When the functions are listed in order from their smallest, to
largest value of a, the order would be;
m( x), g ( x), f ( x)
A.
f ( x), g ( x), m( x)
B.
f ( x), m( x), g ( x)
C.
g ( x), m( x), f ( x)
D.
g x
f x
mx
5
2
4
A quadratic function has x-intercepts at
and . One factor
3
3
of this function must be;
A.
3x  4
B.
2x  3
C.
3x  2
D.
4x  3
5
x
5
10
10
9
8.
The graph shown to the right is of y  ( x  p) 2  q . This
graph must have a y-intercept at;
A.
(0, 25)
B.
(0, 31)
C.
(0, 35)
D.
(0, 41)
8
7
6
5
f x
4
3
2
1
1
1
0
1
2
3
4
5
6
7
8
1
9.
The quadratic function y  k ( x  3) 2  m would have its
vertex at;
(3, m)
A.
(3, m)
B.
(3, k )
C.
( 3, k )
D.
10.
A quadratic function has is given by the equation f ( x)  ( x  k )  m . Which of the following is true?
A.
The function has a minimum at ( k , m)
B.
The function has a maximum at ( k , m)
C.
The function has a minimum at (k ,m)
D.
The function has a maximum at (k ,m)
1
x
8
5
5
4
11.
The graph shown to the right is of f (x) . If g (x ) is a reflection
around the x-axis of f (x) , g (x ) could be given by the function;
A.
g ( x)  ( x  2) 2  3
B.
g ( x)  ( x  2) 2  3
C.
g ( x)  ( x  2) 2  3
D.
g ( x)  ( x  2) 2  3
3
2
1
f x
1
0
1
2
3
4
5
6
7
1
2
3
4
5
5
1
x
7
12.
1
The area of a triangle is given by the formula A  bh , where b is the base length,
2
and h is the height. According to the diagram on the right, with the dimensions of b
and h as given, the base length, in order that the area of the triangle is a maximum,
must be;
A.
12 units2
B.
9 units2
C.
6 units2
D.
3 units2
13.
A quadratic function with a maximum at (1,2) passes through the point (-3,-4). If the function is
given by y  a( x  p) 2  q , then the value of a must be;
A.
–2
B.
2
1
C.
2
1
D.
2
14.
According to the graph shown on the right, with x-intercepts at
(1,0) and (5,0) , and a y-intercept at (0,5) one factor of f (x )
must be;
A.
x5
B.
x 5
C.
x 1
D.
5x  1
5
4
3
2
1
5
f x
10
4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
4
x
10
15.
The graph of f ( x)  a( x  p) 2  q has no x-intercepts. Which of the following must be true?
A.
If a  0 , then q  0
B.
If a  0 , then q  0
C.
If a  0 , then p  0
D.
If a  0 , then p  0
16.
The factored form of a quadratic is given by f ( x)  ( x  a)( x  5) . If f (x) has a y-intercept at (0,10) ,
the value of a must be;
A.
-2
B.
2
C.
-5
D.
5
17.
When the quadratic function y  x 2  4 x  7 is written in the form y  a( x  p)2  q , the value of q must
be;
A.
11
B.
-11
C.
9
D.
-9
18.
Which of the following statements is true about the quadratic function f ( x)  ( x  2)2  q ?
A.
Its maximum is located at (2, q)
B.
Its minimum is located at (2, q)
C.
Its maximum is located at (-2, q)
D.
Its minimum is located at (-2, q)
19.
Two functions are shown on the graph to the right. Both functions
share an axis of symmetry at x  2 , and can be described generally
by the quadratic function y  a( x  p)2  q . For the parameters a, p,
and q, the functions must;
A.
Have none of a, p, or q in common
B.
Have exactly one of a, p, or q in common
C.
Have exactly two of a, p, or q in common
b
D.
Have all three of a, p, and q in common
f x
g x
1
1
0
x
0
1
20.
21.
22.
The diagram to the right shows a rectangle inscribed in a rightangled triangle with a hypotenuse given by the equation of the
line y  2 x  4 , and the other sides bounded by the x and y-axis.
The size of this rectangle changes as the point on the line moves.
Which of the following is true about the area, A, of this rectangle?
A  2 x 2  4 x , and is a maximum when A  4
A.
A  2 x 2  4 x , and is a maximum when A  2
B.
A  2 x 2  4 x , and is a maximum when A  4
C.
A  2 x 2  4 x , and is a maximum when A  2
D.
When the function shown to the right is written in the form
y  ax 2  bx  c , the value of c must be;
A.
–12
B.
4
C.
12
D.
-4
1
A quadratic function has zeros of
and 3. The function must have
2
factors of;
A.
Both 2x  1 and x  3
B.
Both 2x  1 and x  3
C.
Both 2x 1 and x  3
D.
Both 2x 1 and x  3
6
5
4
f x
3
b
2
1
1
0
1
2
3
4
1
x, a
6
4
2
4
f x
3
2
1 0 1
2
4
6
8
10
12
14
16
x
2
3
4
5
6
23.
The graph of f ( x)  a( x  2)2  q holds the property that q  2a , a  0 . Which of the following must be
true?
f ( x ) has two x-intercepts
A.
f ( x ) has only one x-intercept
B.
f ( x ) could must have at least one, but no more than two x-intercepts
C.
f ( x ) must have no x-intercepts
D.
24.
The y-intercept of the quadratic function f ( x)  2( x  k )2  5 can be expressed as;
A.
-5
2 k 2  5
B.
10k 2
C.
D.
2k  5
25.
The quadratic function y = a(x + k)2 - 4 has a vertex at:
A.
B.
C.
D.
26.
A quadratic function with an x-intercept at Ошибка! must have a factor of:
A.
B.
C.
D.
28.
(k, 4)
(-k, 4)
(k, -4)
(-k, -4)
(3x + 4)
(3x - 4)
(4x + 3)
(4x - 3)
y
4
The graph shown to the right is y = a(x - 4)2 - 6.
The value of a must be:
A.
B.
-2
2
C.
Ошибка!
2
-4
-2
y Intercept
(0,2)
2
4
6
8
10
x
-2
-4
D.
Ошибка!
-6
y
29.
A swimming area is roped off on three sides with 100 m
of rope, and the beach on one side. Which of the
following expressions relating area, A to the width of
this swimming area, x, would assist in finding the largest
possible area for this swimming area:
A.
B.
C.
D.
A
A
A
A
=
=
=
=
-8
10
8
6
2(x2 - 50x)
2(x2 + 50x)
-2(x2 - 50x)
-2(x2 + 50x)
4
2
Local Minimum
(6,2)
-4
-2
2
-2
4
6
8
10
x
30.
The graph shown to the right is y = (x - p)2 + q. An ordered pair (12, k) lies on this parabola. k
must be:
A.
B.
C.
D.
31.
Which of the following is a value for k making 3x2 + kx - 6 factorable?
A.
B.
C.
D.
32.
6
66
38
16
18
-19
-7
13
The quadratic function f(x) = -2(x + 3)2 - 7 has a y-intercept of:
A.
B.
C.
D.
(0, -25)
(0, -7)
(0, -14)
(0, -18)
33.
A function f(x) = kx2 - m ,
A.
B.
C.
D.
has its vertex at (k, -m), and opens up if k > 0.
has its vertex at (-k, m), and opens down if k > 0
has its vertex at (0, -m), and opens up if k > 0.
has its vertex at (0, m), and opens down if k > 0.
Numberical Response Questions
1.
When y  3x 2  18x  37 is written in the form y  3( x  3) 2  k , the
value of k must be _____.
10
f x
2.
A function, f (x) has its inverse displayed in the graph to the right.
The number of x-intercepts of the function f (x) must be _____.
3.
The largest value of k making 2 x  kx  6 factorable must be _____.
4.
A model rocket is launched from the roof top of a building, and
it’s flight pattern is modeled by the graph shown to the right,
where f (t ) is the height of the rocket in metres, at time t, given
in seconds. According to the model, the height of the roof top
must be _____.
g x
10
2
10
50
x
20
50
45
40
35
30
f t
25
20
15
5.
The function y  x 2 is transformed four times, as shown below.
Each transformation is independent of any other in this
question.
Transformation #1:
Transformation #2:
Transformation #3:
Transformation #4:
10
5
0
0 1
0
2
3
4
5
t
6
7
8
9 10
10
y  x 2 becomes y  kx2 , where k  1 .
y  x 2 becomes y  kx2 , where 0  k  1.
y  x 2 becomes y  ( x  k ) 2 , where k  0 .
y  x 2 becomes y  ( x  k ) 2 , where k  0 .
Place your answer for each of the following in the answer space for question #5, starting with the first
available box.
The transformation illustrating a horizontal expansion is illustrated in transformation # _____.
The transformation illustrating a movement of the vertex to the right is illustrated in transformation #
_____.
The transformation illustrating a horizontal compression is illustrated in transformation # _____.
The transformation illustrating a movement of the vertex to the left is illustrated in transformation #
_____.
6.
The first two steps shown below are a completion of the square for the quadratic equation
y  2 x 2  12 x  15
y  a( x 2  6 x)  15
y  a( x 2  6 x  k )  15  m
The value of a, k, and m must be _____ (Place your response on the answer sheet in the four boxes
provided for question #1, beginning with the far left box).
7.
An enclosed area is designed using one existing wall, and 72 m of fence material. The maximum
possible area of this enclosure must be _____m2.
8.
The largest value of k making 3x 2  kx  4 factorable must be
_____.
9.
A rock is thrown from the edge of a cliff. A graph showing the
height in metres, h(t ) relative to the time in seconds, t, is shown
to the right. According to the graph, the approximate height of
the rock after 3 seconds is _____.
80
h t
0
80
76
72
68
64
60
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0 1
0
2
3
4
5
t
6
7
8
9 10
10
10.
The functions f (x) and g (x ) are shown to the
right. Both can be represented by the equation
y  a( x  p) 2  q . Only one parameter (that is, a, p
or q) remains unchanged. The value of the
parameter that remains unchanged is _____.
10
9
8
7
6
5
4
3
2
1
10
f x
5
g x
4
3
10
2
11 0 1
2
3
4
5
6
7
8
9
10
5
11.
2
3
4
x
5
5
A student is examining four functions, illustrated in the chart shown below. Each one is missing at least
one parameter.
Function #1
f ( x)  a( x  3)  2 , a  0
2
Function #2
f ( x)  2( x  p)
2
Function #3
Function #4
f ( x)  ax  6 , a  0
f ( x)  2 x 2  bx  7
2
Choose the proper function as numbered above, and respond to each of the following by beginning your
answer in the far left hand box for question #1, and completing your answer in the far right hand box for
question #1. Each function can only be used once in your solution.
The function that must have its minimum on the x-axis is illustrated by function #_____.
The function that has a negative y-intercept is illustrated by function # _____.
The function that has its vertex in the second quadrant is illustrated by function # ____.
The function whose range is restricted by a minimum positive value, zero not included, is illustrated by
function #_____.
500
12.
The graph shown on the right illustrates possible areas,
A, of rectangles having a perimeter of 80 units. If x
represents a side length of the rectangle, a rectangle of
300 units2 would have its longest side _____ units.
450
400
350
300
A x 250
200
150
13.
If f (2)  5 , and the inverse of f is a function, then
f 1 (5) must equal _____.
100
50
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
x
14.
In completing the square of the trinomial f ( x)  2 x 2  12 x  5 , the result is the quadratic function
f ( x)  2( x  p)2  q , with a q value of _____.
15.
Correct to the nearest hundredth, the largest zero of the function y  4 x 2  11x  3 is _____.
16.
f(x) = -3x - 1. If (k, -2) is on f-1(x), then k must be _____.
17.
When y = 2x2 + 8x + 11 is written in the form y = 2(x + 2)2 + k, k must equal _____.
18.
A quadratic function is congruent to y = 2x2, has an axis of symmetry at x = -2, and a minimum value
of 8. When written in the form y = ax2 + bx + c, the values of a, b, and c are (place the values in
successive boxes on your answer sheet for question 3 starting with the far left box and ending with the
far right box).
19.
The function y = 3x2 - 14x - 5 has two x-intercepts. The positive x-intercept is _____.
20.
Two functions are shown to the right. Both can be written in the form y = a(x - p)2 + q. How
many parameters (a, p, and q) must change so f(x) = m(x)?
21.
f(x) = -3x - 1. If (k, -2) is on f-1(x), then k must be _____.
22.
When y = 2x2 + 8x + 11 is written in the form y = 2(x + 2)2 + k, k must equal _____.
23.
A quadratic function is congruent to y = 2x2, has an axis of symmetry at x = -2, and a minimum value
of 8. When written in the form y = ax2 + bx + c, the values of a, b, and c are (place the values in
successive boxes on your answer sheet for question 3 starting with the far left box and ending with the
far right box).
24.
The function y = 3x2 - 14x - 5has two x-intercepts. The positive x-intercept is _____.
25.
Two functions are shown to the right. Both can be written in the form y = a(x - p)2 + q. How
many parameters (a, p, and q) must change so f(x) = m(x)?
26.
The quadratic function y = -a(x - 5)2 + m has a vertex at (x, 2x). The value of m must be _____.
27.
The graph of y  a( x  p) 2  q is shown to the right. The value of a must be _____.
28.
Correct to the nearest tenth Ошибка! has a y-intercept of _____.
30.
A triangle, whose area can be expressed at Ошибка!, where bis the base length, and h is the height, has
a base length of (4 - x) units, and a height of 8x units. The maximum area of this triangle must be _____
units2.
Written Response Questions
1.
A rental car agency has 200 cars. At $36.00 per day, all cars will be rented. However, for each $2.00
increase in price, 5 fewer cars will be rented each day.
a.
Determine an algebraic model representing this problem, where R is the agency’s revenue, and x
is each $2.00 increase in the rental price of a car. Write your final model in the form
R  ax 2  bx  c
b.
“Revenue for this rental agency is maximized when cars are rented for $58.00”. Using your
model from question (a), algebraically prove this statement to be true. In other words, complete
the square with the model you have derived, and make a concluding statement supporting the
maximized value.
c.
4
100001 10
9000
The graph shown to the right is a geometric
representation of the model for this
problem.
8000
7000
If the manager of the car rental agency
informs you that his “break even point” is
$4000.00 (in other words, the rental agency
cannot afford to have revenues less than
that amount), explain to the manager how
he can use the graph of this model to assist
in decision-making. Use any necessary
diagrams of your own to support your
answer.
6000
R x
5000
4000
3000
2000
1000
0
0 4
8 12 16 20 24 28 32 36 40
0
2.
x
State the following equations in the form y  a( x  p)  q using
the given information.
a.
b.
The parabola has it’s axis of symmetry at x = 3, and passes
through the points (-1,-27) and (2,3).
The parabola matches the graph shown to the right.
6
6
5
4
3
f x
2
1
2
3.
40
2
A truck is traveling down the highway, and approaches a parabolic
tunnel similar to the diagram shown to the right. The tunnel
1 2
x  m . Each unit on the graph’s grid
2
represents 1 metre, and the truck must stay in his driving lane (e.g.
to the right of the y-axis).
satisfies the equation y 
a.
b.
c.
Determine the value of m, and give support for your
answer.
Explain why a truck 3 m wide, and 5 m high can travel
through this tunnel.
If the truck in this problem is 2.5 m wide, determine the
maximum height possible in order that it can travel through
the tunnel, correct to the nearest tenth of a metre.
1
0
1
2
3
4
5
6
1
2
2
2
x
6
12
11
10
9
8
7
6
5
4
3
2
1
12
f x
0
6 5 4 3 2 1 0 1 2 3 4 5 6
6
x
6
4.
A cruise boat is chartered for an excursion. With 100 people on the boat, each ticket will cost $30.00.
For each additional 10 people that sign up for the cruise, the ticket price will be reduced by $1.00.
a.
Determine an algebraic model representing this problem, where R is the cruise boat’s revenue,
and x is each $1.00 decrease in the cost of a ticket. Write your final model in the form
R  ax 2  bx  c
b.
The cruise operator determines that the maximum revenue she can generate is $4000, based on a
ticket price of $20.00. Show this must be true by using your answer from (a), and writing it in
the form y  a( x  p)2  q through completion of the square, and explain the significance of the
ordered pair (p, q) to this problem.
c.
The graph shown to the right is a geometric
representation of the model for this
problem.
5000
4500
4000
d.
Federal regulators inform the cruise
operator that she cannot have more than
160 people on her boat at one time.
Explain how this graph can be used to
determine the maximum revenue the cruise
operator can generate given this
information?
3500
3000
R x 2500
2000
1500
1000
500
0 2 4 6 8 10 1214 1618 2022 2426 28 30
x
Design of Arch on 4 m Pillars
An engineer was asked to sketch a tunnel using a parabolic arch
design which will rest on two 4 m pillars. The engineer
sketches a diagram representing the arch design. That sketch is
shown to the right, with all units in metres.
a.
Given the arch has a maximum height of 9 m, determine
the equation of the parabola used to make the tunnel,
and express your answer in the form y  a( x  p)2  q .
10
9
8
f x
Height (m)
5.
b
7
6
5
b
4
3
2
1
c.
Explain why the parameter a in your answer from
question (a) is such that a  0 .
A two lane road will pass under this tunnel. Due to
increased cost in building the road, the municipality has
decided that the tunnel must be completely redesigned
so it now only rests on 3 m pillars, and the parabolic
arch cannot be as high. The result is shown in the
diagram to the right. Given the equation of the new
parabolic arch is f ( x)  0.04( x  5)2  4 , explain how
to determine the height of the truck shown in the
diagram, and express the resulting height to the nearest
hundredth.
0 1
2
3
4
5
6
7
8
9 10
x, a , c
Width (m)
Redesign of Arch on 3 m Pillars
6
f x
5
b
Height (m)
b.
b
h
h
4
3
2
n
1
0 1
2
3
4
5
6
7
8
9 10
x, a , c , g , i, k
Width (m)
6.
Answer the following regarding a parabola which could be represented in the form y  a( x  p)2  q .
a.
If the axis of symmetry of this parabola is x  3 , which parameter, a, p or q can be determined
from that information?
b.
If the parabola described in (a) has a minimum at y  2 , what do we know about the value of
the parameter a?
c.
If the parabola described in (a) and (b) has a y-intercept of (0,16), determine its equation in the
from y  a( x  p)2  q .
d.
Only the information given in part (a) is relevant to this question. If a second parabola, different
than the one described in part (c) were to pass through the points (-1, -12) and (2, 3), determine
its equation in the form y  a( x  p)2  q .
7.
Using the quadratic function y = 3x2 - 7x + 4,
a.
b.
c.
8.
9.
State the location of the y-intercept.
Find the values of the two x-intercepts by factoring. Show all steps.
Find the value of the vertex by writing the quadratic in the form y = a(x - p)2 + q.
Find each equation of the two quadratics described below. Express your answer in the form
y = a(x - p)2 + q.
a.
Matches the graph shown to the right.
b.
Has an axis of symmetry at x = -2, and passes through the points (-1, 4) and (1,-2).
Over a period of four games, a sports franchise experimented with the prices of tickets for their nonseason ticket holding fans. The table below shows the number of fans that purchased tickets, and the
price they paid over those four games.
Game
1
2
3
4
Price
$30
$28
$26
$24
Tickets Purchased
2000
2400
2800
3200
a.
Develop an expression relating revenue, R with each increase in price illustrated by the data, x.
Write your expression in the form y = ax2 + bx + c.
b.
What should this sports franchise charge for these tickets in order to maximize the amount of
revenue?
c.
At what two ticket prices could they generate revenue of $60 000?