1. MTH-4108 C
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 = 3.2x2 + 4x
y
−b −∆
 , 
 2a 4 a 
Vertex:  − 4 − 16 
 , 
 6.4 12.8 
( − 0.625,−1.25)
x −b± ∆
2a
Zeros: − 4 ± 16
6.4
{ − 1.25,0}
y = 3.2(0) + 4(0)
y-intercept:
y=0
b) y = ¼x2 – 3x +1
y
Vertex:
−b −∆
 , 
 2a 4a 
 3 − ( 9 − 4( 0.25)(1) ) 
 , 
 0.5 1 
( 6,−8)
x
−b± ∆
2a
Zeros: 3 ± 8
0.5
{ 0.343,11.65}
y-intercept:
y = 0.25(0) − 3(0) + 1
y =1

2. c) y = –3x2 + x – 2 Vertex:
y −b −∆
 , 
 2a 4a 
 − 1 − (1 − 4( − 3)( − 2 ) ) 
 , 
−6 − 12 
( 0.166,−1.9166)
−b± ∆
2a
x Zeros: − 1 ± − 23
−6
{ ∅}
y-intercept:
y = −3(0) + (0) − 2
y = −2
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2. Solve the following equations by factoring:
a) 8x2 – 2x = 0 2 x(4 x − 1) = 0
2x ⇒ 2x = 0 ⇒ x = 0
( 4 x − 1) ⇒ 4 x = 1 ⇒ x = 1 4
b) 3/5x2 – 2/5x + 1/5= 0 ( )
1 3x 2 − 2 x − 1 = 0
5
3x 2 − 3x + 1x − 1 = 0
3x( x − 1) + 1( x − 1)
(3x + 1)( x − 1) = 0
( 3x + 1) ⇒ 3x = −1 ⇒ x = − 13
( x − 1) ⇒ x = 1 /5
3. Identify the true statement(s) below which pertain to quadratic functions in the
form: y = ax2 + bx + c
a) If a = 0, it means that the function is no longer quadratic. TRUE
____________
FALSE
b) If c = 0, it means that the vertex is at (0,0). ____________
TRUE
c) If b = 0, it means that the function is situated on the y-axis. ____________
d) If a > 0, it means that the function opens upward. TRUE
____________
e) If c > 0, it means that the function is above the x-axis. FALSE
____________
/5

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.
5
a) /3x2 – 2/5x + 1/2= 0 Zeros:
−b± ∆
2a
0.4 ± ( 0.16 − 4(1.667 )( 0.5) )
1
0.4 ± − 3.1733
1
{ ∅}
b) 2x – 3x2 +2 = 0
−b± ∆
2a
Zeros: − 2 ± ( 4 − 4( − 3)( 2 ) )
−6
{ − 0.5485,1.2152}
/10
5. Every month, Irene sells 6 dozen roses that she grows in her garden for $20 per
dozen. For every additional dozen roses she grows, she can reduce her price by $2
per dozen. How many dozen roses should she grow every month in order to
maximize her total sales? Complete the following table and write the equation that
can be used to solve the problem. Your equation should be in the form y = ax2 +
bx + c
Number of Total number of Selling Price ($) Total Sales ($)
additional dozens of roses y
dozens of roses
x
0 6 20 6 × 20 = 120
1 6+1=7 20 – 2 = 18 7 × 18 = 126
2 6+2=8 20 – 2(2) = 16 8 × 16 = 128
3 6+3=9 20 – 2(3) = 14 9 × 14 = 126
x 6+x 20 – 2x (20 – 2x)( 6 + x)
Write the equation in the form ax2 + bx + c which illustrates this situation.
( 20 − 2 x )( 6 + x )
120 + 20 x − 12 x − 2 x 2
y = −2 x 2 + 8 x + 120
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4. 6. Answer the following questions using the graph below:
y
x
True or False?
TRUE
a) Point (-2,0) is a zero. ____________
TRUE
b) Point (0,-1) is a vertex. ____________
FALSE
c) The equation for the axis of symmetry is: y = 0. ____________
TRUE
d) Point (-0,-1) is a minimum. ____________
FALSE
e) Point (2,0) is the y-intercept. ____________
/5
7. Without calculating, write the equation in the form ax2 + bx + c which illustrates
the following situation:
A mother’s present age is triple her son’s age. In 15 years, the product of their
ages will be 60 times her son’s age then.
Let x = Son’s age
Let 3x = Mother’s age
( x + 15)( 3x + 15) = 60( x + 15)
3 x 2 + 15 x + 45 x + 225 = 60 x + 900
3 x 2 − 675 = 0
/5

5. 8. The hypotenuse of a right triangle is 10 cm. Find the length of the other two sides
if one side is 2 cm longer than the other, using the Pythagorean Theorem.
Let x = small side
Let 2 + x = LARGE side
x 2 + ( x + 2) 2 = 10 2
x 2 + 2 x − 48 = 0
x 2 + x 2 + 4 x + 4 = 100
Zeros: ( x + 8)( x − 6) = 0
2 x 2 + 4 x − 96 = 0
{ − 8,6}
x 2 + 2 x − 48 = 0
2 + (6) = 8
ANS: The two sides measure 6 cm and 8 cm
/5
9. A School principal paid $180 for a certain number of desks. If the desks cost 3
dollars less, she could have bought 5 more desks for the same price. How many
desks did the principal order?
Let x = number of desks ordered
180 180
−3=
x x+5
180 − 3 x 180
= x 2 + 5 x − 300 = 0
x x+5
180 x + 900 − 3 x 2 − 15 x = 180 x Zeros: ( x − 15)( x + 20)
− 3 x 2 − 15 x + 900 = 0
{ − 20,15}
x 2 + 5 x − 300 = 0
ANS: The principal ordered 15 desks.
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10. The function h = 24t – 6t2 represents the height (h) in centimeters reached by a
champion jumping frog after t seconds. Use the method of your choice to find the
frog’s maximum height.
Round your answers to the nearest hundredth.
 − b − ∆   − 24 − ( 576 ) 
Vertex:  , ⇒ ,  ⇒ ( 2,24 )
 2a 4a   − 12 − 24 
/5

6. 11. Tarzan swings on a vine from a tall tree, across a river, and rises upwards to
another tree in a parabolic arc. The equation f = 0.04x2 – 5x + 28 describes his
motion, withboth the fall (f) and distance (x) between the two trees measured in
feet. Use the method of your choice to answer the following questions:
a) What is the distance between the two trees?
−b± ∆ 5 ± (−5) 2 − 4( 0.04)( 28) 5 ± 20.52
⇒ ⇒
2a 2(0.04) 0.08
Zeros:
{ 5.876,119.123}
119.123 − 5.876 = 113.2475
b) What is the measure of the vertical drop?
 − b − ∆   5 − ( 20.52 ) 
Vertex:  , ⇒
 ,  ⇒ ( 62.5,−128.25)

 2a 4a   0.08 4(0.04) 
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