Math 20-1 – Pre-calculus 11

Quadratic Functions Review

Multiple Choice Questions

1. The quadratic function mxy 2 would have its vertex at;

A. ),1( m

B. ),0( m

C. ),1( m

D. ),0( m

2. A quadratic function has a range such that Ryy ,4 , with a domain Rx . The number of x-

intercepts of this function must be;

A. 0

B. 1

C. 2

D. 3

3. The function and

A. The same range.

B. The same domain.

C. A different vertex.

D. A maximum at y = -2.

4. A rectangle has a width of x2 , and a length of x4 . The maximum area of this rectangle must be;

A. 4 units2

B. 6 units2

C. 8 units2

D. 16 units2

5. A quadratic function passes through the point (-2,-9) and has a maximum at -5 when x = -1. When

written in the form qpxay 2)( , the value of a must be;

A. 4

B. -4

C. 2

D. -2

6. Three functions, all of the form 2axy , have a = -4, -1, 2 and 6. Which of the following is true?

A. The all contain the same y-intercept only.

B. They all contain the same x-intercept only.

C. They all share both the same x and y intercept.

D. They don’t contain either the same x or y intercept.

7. A quadratic function has x-intercepts at 3

4and

3

2. One factor of this function must be;

A. 43x

B. 32x

C. 23x

D. 34x

8. The graph shown to the right is of 2( 5) 6y x . When

the value of a is changed from 1 to -2, the value of the y-

intercept changes to

A. -44

B. -62

C. 44

D. 62

9. The quadratic function mxky 2)3( would have its

vertex at;

A. ),3( m

B. ),3( m

C. ),3( k

D. ),3( k

10. A quadratic function has is given by the equation mkxxf )()( . Which of the following is true?

A. The function has a minimum at ),( mk

B. The function has a maximum at ),( mk

C. The function has a minimum at ),( mk

D. The function has a maximum at ),( mk

11. The graph shown to the right is of )(xf . If )(xg is a reflection

around the x-axis of )(xf , )(xg could be given by the function;

A. 3)2()( 2xxg

B. 3)2()( 2xxg

C. 3)2()( 2xxg

D. 3)2()( 2xxg

1 0 1 2 3 4 5 6 7 81

1

2

3

4

5

6

7

8

9

1010

1

f x

81 x

1 0 1 2 3 4 5 6 7

5

4

3

2

1

1

2

3

4

55

5

f x

71 x

12. The area of a triangle is given by the formula bhA2

1, where b is the base length,

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 )2,1( passes through the point (-3,-4). If the function is

given by qpxay 2)( , then the value of a must be;

A. –2

B. 2

C. 2

1

D. 2

1

14. According to the graph shown on the right, with x-intercepts at

)0,1( and )0,5( , and a y-intercept at )5,0( one factor of )(xf

must be;

A. 5x

B. 5x

C. 1x

D. 15x

15. The graph of qpxaxf 2)()( has no x-intercepts. Which of the following must be true?

A. If 0a , then 0q

B. If 0a , then 0q

C. If 0a , then 0p

D. If 0a , then 0p

16. The factored form of a quadratic is given by )5)(()( xaxxf . If )(xf has a y-intercept at )10,0( ,

the value of a must be;

A. -2

B. 2

C. -5

D. 5

4 3 2 1 0 1 2 3 4 5 6 7 8 9 10

109

8

7

6

5

4

3

2

1

1

2

3

4

55

10

f x

104 x

17. When the quadratic function 2 4 7y x x is written in the form 2( )y a x p 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 2( ) ( 2)f x x 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 2x , and can be described generally

by the quadratic function 2( )y a x p 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

D. Have all three of a, p, and q in common

20. The diagram to the right shows a rectangle inscribed in a right-

angled triangle with a hypotenuse given by the equation of the line

2 4y x , 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. 22 4A x x , and is a maximum when 4A

B. 22 4A x x , and is a maximum when 2A

C. 22 4A x x , and is a maximum when 4A

D. 22 4A x x , and is a maximum when 2A

21. When the function shown to the right is written in the form 2y ax bx c , the value of c must be;

A. –12

B. 4

C. 12

D. -4

22. A quadratic function has zeros of 1

2 and 3. The function must have

factors of;

A. Both 2 1x and 3x

B. Both 2 1x and 3x

C. Both 2 1x and 3x

D. Both 2 1x and 3x

f x

g x

x

b

1

1

0

0

1

1 0 1 2 3 4

1

1

2

3

4

5

6

f x

b

,x a

4 3 2 1 0 1 2 3 4 5 6

16

14

12

10

8

6

4

2

2

4

6

f x

x

23. The graph of 2( ) ( 2)f x a x q holds the property that 2q a , 0a . Which of the following must be

true?

A. ( )f x has two x-intercepts

B. ( )f x has only one x-intercept

C. ( )f x could must have at least one, but no more than two x-intercepts

D. ( )f x must have no x-intercepts

24. The y-intercept of the quadratic function 2( ) 2( ) 5f x x k can be expressed as;

A. -5

B. 22 5k

C. 210k

D. 2 5k

25. The quadratic function y = a(x + k)2 - 4 has a vertex at:

A. (k, 4)

B. (-k, 4)

C. (k, -4)

D. (-k, -4)

26. A quadratic function with an x-intercept at -4

3 must have a factor of:

A. (3x + 4)

B. (3x - 4)

C. (4x + 3)

D. (4x - 3)

27. The graph shown to the right is y = a(x - 4)2 - 6.

The value of a must be:

A. -2

B. 2

C. -1

2

D. 1

2

28. 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. A = 2(x2 - 50x)

B. A = 2(x2 + 50x)

C. A = -2(x2 - 50x)

D. A = -2(x2 + 50x)

x-4 -2 2 4 6 8 10

y

-8

-6

-4

-2

2

4

y Intercept( 0 , 2 )

29. 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. 6

B. 66

C. 38

D. 16

30. Which of the following is a value for k making

3x2 + kx - 6 factorable?

A. 18

B. -19

C. -7

D. 13

31. The quadratic function f(x) = -2(x + 3)2 - 7 has a y-intercept of:

A. (0, -25)

B. (0, -7)

C. (0, -14)

D. (0, -18)

32. The root of is derived from a factor of

A. x – 2

B. x + 2

C. 2x – 1

D. 2x + 1

33. When completely factored, is a product of

A. Two factors

B. Three factors

C. Four factors

D. Five factors

34. A polynomial expression is given by . One of its factors is

A. t + 2

B. t – 4

C. 3t - 4

D. 3t + 1

35. In order for to be a perfect square trinomial, m must be

A. ±144

B. ±12

C. ±48

D. ±24

Local Minimum( 6 , 2 )

x-4 -2 2 4 6 8 10

y

-2

2

4

6

8

10

36. One root of the equation is

A.

B.

C.

D.

37. The zeros of a quadratic function are 4 and -3. If the graph of the function has a y-intercept of 48, the

quadratic must be;

A.

B.

C.

D.

38. Which of the following statements is true regarding

A. the quadratic function shown on the right?

B.

C.

D.

E.

39. A function f(x) = kx2 - m ,

A. has its vertex at (k, -m), and opens up if k > 0.

B. has its vertex at (-k, m), and opens down if k > 0

C. has its vertex at (0, -m), and opens up if k > 0.

D. has its vertex at (0, m), and opens down if k > 0

40. The smallest x-intercept of a quadratic function is located at x = k, k>0. If the axis of symmetry is

located at x = 8, then the other x intercept must be located at

A. 4 – k

B. 10 – k

C. 12 – k

D. 16 - k

x-4 -2 2 4 6 8 10

y

-2

2

4

6

8

10

Numerical Response Questions

1. When 37183 2 xxy is written in the form kxy 2)3(3 , the value of k must be _____.

2. The value of k making 24 24x x k a perfect square trinomial is _____.

3. 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 )(tf 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 _____m.

4. The function 2xy is transformed four times, as shown below. Each transformation is independent of

any other in this question.

Transformation #1: 2xy becomes 2kxy , where 1k .

Transformation #2: 2xy becomes 2kxy , where 10 k .

Transformation #3: 2xy becomes 2)( kxy , where 0k .

Transformation #4: 2xy becomes 2)( kxy , where 0k .

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 #

_____.

5. The first two steps shown below are a completion of the square for the quadratic equation

15122 2 xxy

15)6( 2 xxay

mkxxay 15)6( 2

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).

6. 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.

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

35

40

45

5050

0

f t

100 t

7. The largest value of k making 43 2 kxx factorable must be _____.

8. A rock is thrown from the edge of a cliff. A graph showing the

height in metres, )(th 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 _____m.

9. The function 2( ) ( )f x a x p q containing the ordered pair (-2,-12) is reflected about the x-axis. The

reflection contains the ordered pair (a, b). The value of a b is _____.

10. A student is examining four functions, illustrated in the chart shown below. Each one is missing at least

one parameter.

Function #1 Function #2 Function #3 Function #4

2( ) ( 3) 2f x a x , 0a 2( ) 2( )f x x p 2( ) 6f x ax , 0a 2( ) 2 7f x x bx

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 y-intercept of -7 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 #_____.

11. 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.

0 1 2 3 4 5 6 7 8 9 10

48

121620242832364044485256606468727680

80

0

h t

100 t

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

50

100

150

200

250

300

350

400

450

500

A x

x

12. In completing the square of the trinomial 2( ) 2 12 5f x x x , the result is the quadratic function 2( ) 2( )f x x p q , with a q value of _____.

13. The sum of the two zeros from the function 23 5 2y x x correct to the nearest hundredth is _____.

14. When y = 2x2 + 8x + 11 is written in the form y = 2(x + 2)2 + k, k must equal _____.

15. 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).

16. The function y = 3x2 - 14x - 5 has two x-intercepts. The positive x-intercept is _____.

17. 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 the two functions are congruent?

18. The quadratic function y = -a(x - 5)2 + m has a vertex at (x, 2x). The value of m must be _____.

19. The graph of qpxay 2)( is shown to the right. The value of a

must be _____.

20. Correct to the nearest tenth f(x) = 1

2(x - 5)(x - 3) has a y-

intercept of _____.

21. A triangle’s area can be expressed as A = 1

2bh, where b is the base

length, and h is the height. If the base length is (4 - x) units, and the height is 8x units, the maximum

area of this triangle must be _____ units2.

x-2 2 4 6 8

y

-4

-2

2

4

x-2 2 4 6 8

y

-8

-6

-4

-2

2

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

cbxaxR 2

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. The graph shown to the right is a geometric

representation of the model for this

problem.

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.

2. State the following equations in the form qpxay 2)( using the given information.

a. The parabola has it’s axis of symmetry at x = 3, and passes through the points (-1,-27) and (2,3).

b. The parabola matches the graph shown to the right.

0 4 8 12 16 20 24 28 32 36 40

1000

2000

3000

4000

5000

6000

7000

8000

9000

1 104

10000

0

R x

400 x

2 1 0 1 2 3 4 5 6

2

1

1

2

3

4

5

66

2

f x

62 x

3. A truck is traveling down the highway, and approaches a parabolic tunnel similar to the diagram shown

to the right. The tunnel satisfies the equation mxy 2

2

1. Each

unit on the graph’s grid represents 1 metre, and the truck must stay

in his driving lane (e.g. to the right of the y-axis).

a. Determine the value of m, and give support for your

answer.

b. Explain why a truck 3 m wide, and 5 m high can travel

through this tunnel.

c. 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.

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

cbxaxR 2

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 2( )y a x p 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. 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?

6 5 4 3 2 1 0 1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

10

11

1212

0

f x

66 x

0 2 4 6 8 1012141618202224262830

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

R x

x

5. 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 2( )y a x p q .

b. Explain why the parameter a in your answer from

question (a) is such that 0a .

c. 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 2( ) 0.04( 5) 4f x x , explain how

to determine the height of the truck shown in the

diagram, and express the resulting height to the nearest

hundredth.

6. Answer the following regarding a parabola which could be represented in the form 2( )y a x p q .

a. If the axis of symmetry of this parabola is 3x , which parameter, a, p or q can be determined

from that information?

b. If the parabola described in (a) has a minimum at 2y , 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

form 2( )y a x p 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 2( )y a x p q .

7. Using the quadratic function y = 3x2 - 7x + 4,

a. State the location of the y-intercept.

b. Find the values of the two x-intercepts by factoring. Show all steps.

c. Find the value of the vertex by writing the quadratic in the form y = a(x - p)2 + q.

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6Redesign of Arch on 3 m Pillars

Width (m)

Heig

ht

(m)

f x

b

b

h

h

n

,,,,,x a c g i k

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10Design of Arch on 4 m Pillars

Width (m)

Heig

ht

(m)

f x

b

b

,,x a c

8. 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).

9. Over a period of four games, a sports franchise experimented with the prices of tickets for their non-

season ticket holding fans. The table below shows the number of fans that purchased tickets, and the

price they paid over those four games.

Game Price Tickets Purchased

1 $30 2000

2 $28 2400

3 $26 2800

4 $24 3200

a. Develop an expression relating revenue, R with each decrease 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 ticket price could they generate revenue of $60 000?

x-2 2 4 6 8

y

-5

5

10