1. MTH-4108 A
Quadratic Functions
ANSWER KEY
1. Graph the following equations. Be sure to include the coordinates of at least 5
points, including the vertex, the zeros (if any) and y-intercept.
a) y = 0.4x2 – 3x −b −∆
 , 
y  2a 4 a 
Vertex:  3 − 9 
 , 
 0.8 1.6 
( 3.75,−5.62)
−b± ∆
2a
Zeros: 3 ± 9
x
0.8
{ 0,7.5}
y = 0.4(0) − 3(0)
y-intercept:
y=0
b) y = –x2 + 5x – 6.25
y
Vertex:
−b −∆
 , 
 2a 4a 
 − 5 − ( 25 − 4( − 1)( − 6.25) ) 
 , 
−2 −4 
( 2.5,0)
x
−b± ∆
2a
Zeros: − 5 ± 0
−2
{ 2.5,0}
y-intercept:
y = −1(0) + 5(0) − 6.25
y = −6.25

2. c) y = 2x2 + 4x + 3 Vertex:
y −b −∆
 , 
 2a 4a 
 − 4 − (16 − 4( 2)( 3) ) 
 , 
 4 8 
( − 1,1)
−b± ∆
2a
x Zeros: − 4 ± − 8
4
{ ∅}
y-intercept:
y = 2(0) + 4(0) + 3
y =3
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2. Solve the following equations by factoring:
a) 25x2 – 1 = 0 ( 5x − 1) ⇒ 5 x = 1 ⇒ x = 5
1
( 5x + 1) ⇒ 5 x = −1 ⇒ x = − 1 5
2 x 2 + 8 x − 5 x − 20 = 0
b) 2x2 + 3x – 20 = 0 2 x( x + 4 ) − 5( x + 4)
( 2 x − 5) ⇒ 2 x = 5 ⇒ x = 5 2 /5
( x + 4 ) ⇒ x = −4
3. CIRCLE the true statement(s) below:
a) If ∆ = 0 for a quadratic equation, it means that there are 2 solutions with
one solution equal to 0.
b) If the discriminant is less than 0, it means that there are no real solutions
for this equation.
c) If an equation has two zeros which are equal to each other it means that ∆ = 0
d) If an equation has one negative solution, it is impossible that the other
solution is positive if ∆ > 0.
e) If the discriminant is equal to zero then the equation has no solution.
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3. 4. Solve the following equations using the quadratic formula. Clearly indicate the
value of ∆ and round your answers to the nearest thousandth when necessary.
a) 0.5x2 + 4x – 5 = 0 −b± ∆
2a
Zeros: − 4 ± (16 − 4( 0.5)( − 5) )
1
{ − 9.099,1.099}
b) -x2 – x – 11.25 = 0 −b± ∆
3
2a
Zeros: 1 ± (1 − 4(− 13 )( − 11.25) )
−2
3
{ ∅}
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5. A professional hockey player scores 30 goals in one season and earns $1 800 000,
for an average salary of $60 000 per goal. His contract states that for each
additional goal, his average salary will increase by $100. The table below
illustrates his salary as a function of how many goals he scores.
No. of goals Number of Average salary per goal Total salary
scored after 30 total goals
0 30 $60 000 $1 800 000
1 30 + 1 = 31 60 000 + (100× 1) = $60 100 $1 863 100
2 30 + 2 = 32 60 000 + (100× 2) = $60 200 $1 926 400
x 30 + x 60 000 + (100× x) (30 + x)(60000 + 100x)
Write the equation in the form ax2 + bx + c which illustrates this situation.
( 30 + x )( 60000 + 100 x )
1800000 + 3000 x + 60000 x + 100 x 2
y = 100 x 2 + 63000 x + 1800000
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4. 6. Answer the following questions using the graph below:
0.5x2 – x – 4 = y
y
x
Vertex:
a) What are the coordinates of the vertex?
 − b − ∆   1 − (1 − 4( 0.5)( − 4 ) ) 
 , ⇒ ,  ⇒ (1,−4.5)
 2a 4a   1 2 
b) What is the equation of the axis of symmetry?
x=1
c) Is there a maximum or a minimum?
Minimum of -4.5
d) What are the zeros?
−b± ∆ 1± 9
Zeros: ⇒ ⇒ { − 2,4}
2a 1
e) What is the y-intercept?
y-intercept: y = 0.5(0) − (0) − 4 ⇒ y = −4
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7. Without calculating, write the equation in the form ax2 + bx + c which illustrates
the following situation:
The sum of two numbers is 100 and their product is 2 356.
Let x = 1st number
Let 100 – x = 2nd number
x(100 − x ) = 2356
100 x − x 2 = 2356
x 2 − 100 x + 2356 = 0
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5. 8. The small base of a trapezoid measures double the height. The large base
measures 3 metres more than the small base and the area of the trapezoid is equal
to 123.75 metres squared. What are the measurements of the small base, large
base and height of this trapezoid?
Let x = height
Let 2x = small base
Let 2x + 3 = large base
x[ 2 x + ( 2 x + 3) ]
= 123.75 −b± ∆
2
2a
x[ 4 x + 3]
= 123.75 − 3 ± 9 − 4( 4 )( − 247.5)
2
Zeros: 8
4 x 2 + 3 x 123.75
= − 3 ± 3969
2 1
4 x + 3 x = 247.5
2 8
{ − 8.25,7.5}
4 x 2 + 3 x − 247.5 = 0
/5
9. A sports store sells a certain number of bicycles at the regular price and receives
$17 500 profit. The following week the bikes go on sale for $150 less each
bicycle, and the store sells 15 more bicycles for the same total profit as the
previous week. What is the regular price of one bicycle?
Let x = price of one bicycle
17500 17500
+ 15 = −b± ∆
x x − 150
17500 + 15 x 17500 2a
=
x x − 150 150 ± 22500 − 4(1)( − 175000 )
17500 x − 2625000 + 15 x 2 − 2250 x = 17500 x Zeros: 2
150 ± 850
15 x 2 − 2250 x − 2625000 = 0
2
{ − 350,500}
x 2 − 150 x − 175000 = 0
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6. 10. We throw a tennis ball off the school roof. The equation of the height (y) of the
ball in metres is: y = 10 + 8.75t – 5t2, where t represents the time in minutes.
a) What is the maximum height obtained by the ball?
Vertex:
 − b − ∆   − 8.75 − ( 76.5625 − 4( − 5)(10 ) ) 
 , ⇒ ,  ⇒ ( 0.875,13.8)
 2a 4a   − 10 − 20 
b) After how many minutes is the maximum height obtained?
Vertex:
 − b − ∆   − 8.75 − ( 76.5625 − 4( − 5)(10 ) ) 
 , ⇒ ,  ⇒ ( 0.875,13.8)
 2a 4a   − 10 − 20 
c) If the school is 7m high, after how many minutes does the ball land?
7 = −5t 2 + 8.75t + 10
y-coordinate:
0 = −5t 2 + 8.75t + 3
−b± ∆ − 8.75 ± 8.752 − 4( − 5)( 3) − 8.75 ± 136.5625
⇒ ⇒
Zeros: 2a − 10 − 10
{ − 0.294,2.044}
Round your answers to the nearest hundredth.
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