MPM 2DI Quadratic Relations Assignment Solutions
1.
a) Relation A = {(3, 2), (5, 6), (6, 8), (3, -2), (6, -4)}
i) State the domain and range
Domain: {3,5,6}
Range: {2,6,8,-2,-4}
ii) Is this relation a function? Explain.
No, it is not a function. There are x values that
correspond to more than one y value.
b) Relation B = {(7, 1), (5, 1), (-4, 1), (-1, 1)}
i) State the domain and range
Domain: {7,5,-4,-1}
Range: {1}
ii) Is this relation a function? Explain.
Yes, it is a function. Every x value corresponds to
exactly one value corresponds to exactly one y value.
2
2
c) Is this relation x + y = 31 a function? Explain.
No, it is not a function. It is a circle so it will fail the vertical line test.
d) Is this relation y = "2x + 1 a function? Explain.
! Yes, it is a function. It is a line (that is not vertical) so it will pass the vertical line test. Each x value maps to only
one value maps to only one y value.
2.
A soccer!ball is kicked from the ground level. When it has traveled 35 m horizontally, it reaches its maximum height of 25
m. The soccer ball lands on the ground 70 m from where it was kicked.
2
a) Model this situation with a relation in the form y = a ( x ! h) + k .
Vertex (35,25)
There are two other known points: (0,0) and (70,0). Sub in either one.
y = a(x " 35) 2 + 25
0 = a(0 " 35) 2 + 25
0 = 1225a + 25
"25 = 1225a
"0.0204 # a
y = "0.0204(x " 35) 2 + 25 Where y is the height and x is the horizontal distance.
b) What is the soccer ball’s height when it is 50 m from where it was kicked?
Sub in x=50
! y = "0.0204(50 " 35) 2 + 25
!
y = "0.0204(15) 2 + 25
y = "0.0204(225) + 25
y = "4.59 + 25
y = 20.41
The soccer ball’s height is 20.41 m when it is 50 m from where it was kicked.
3.
A football is punted into the air. Its height h, in metres, after t seconds is
! the height of the ball when it was kicked?
a) What was
h = "4.9(0 " 2.4) 2 + 29
h = "4.9(5.76) + 29
h = "28.224 + 29
h = 0.776
h = "4.9(t " 2.4) 2 + 29.
!
The ball was kicked from a height of 0.776 m
b) What was the maximum height of the ball?
The maximum height of the ball is 29 m.
c) How high
! was the ball after 2 s? Was it going up or down at that time? Justify your answer.
h = "4.9(2 " 2.4) 2 + 29
h = "4.9("0.4) 2 + 29
h = "4.9(0.16) + 29
h = "0.784 + 29
h = 28.216
The ball was 28.216 m above the ground after 2 s. The ball was going up because it had not yet reached its highest
point (it reaches the highest point after 2.4 s)
d) When !
does the ball hit the ground?
0 = "4.9(t " 2.4) 2 + 29.
"29 = "4.9(t " 2.4) 2
5.9183673 = (t " 2.4) 2
2.4327695 = t " 2.4
4.83277 = t
The ball hits the ground at approximately 4.83 seconds.
4.
If a pistol bullet is fired vertically at an initial speed of 100 m/s, the height in metres after t seconds is given by
h = 100t !!5t 2 . Find the maximum height attained by the bullet.
h = "5t 2 + 100t
h = "5(t 2 " 20t)
h = "5(t 2 " 20t + 100 "100)
h = "5(t 2 " 20t + 100) + 500
h = "5(t "10) 2 + 500
The maximum height attained by the bullet is 500 m.
5.
An architect has designed a modern building that is to be supported by a steel arch shaped like a parabola. This parabola
2
!
can be modelled
by the relation y = "0.025x + 2x , where y represents the height of the arch and x represents the
distance along the base, both in metres.
a) Convert the relation to vertex form.
y = "0.025x 2 + 2x
!y = "0.025(x 2 " 80x)
y = "0.025(x 2 " 80x + 1600 "1600)
y = "0.025(x 2 " 80x + 1600) + 40
y = "0.025(x " 40) 2 + 40
b) What is the highest point on the parabolic arch?
The highest point on the arch is 40 metres (and it happens 40m from one end at the base)
c) What is the width of the parabolic arch at its base?
! Since the vertex occurs in the middle of the base and the x value of the vertex is 40 m, the whole base is
80 m wide.
6.
Hermione’s mother owns a manufacturing company that produces key rings. Last year, she collected data about the
number of key rings produced per day and the corresponding profit. The data can be modelled by the relation
, where P is the profit in thousands of dollars and k is the number of key rings in thousands.
a) How many key rings must be produced for the maximum profit?
P = "2k 2 + 12k "10
P = "2(k 2 " 6k) "10
P = "2(k 2 " 6k + 9 " 9) "10
P = "2(k 2 " 6k + 9) + 18 "10
P = "2(k " 3) 2 + 8
So k must be 3 to produce the maximum profit. Which means 3000 key rings must be produced.
b) What is the maximum profit?
The maximum value of P is 8, so the maximum profit is $8000.
c) Sketch this relation.
7.
Hiroshi is trying out for the position of kicker on the football team. He wants to know at what angle he should kick the ball
for maximum distance. He has used a machine that kicks footballs with constant velocity but at varying angles. Hiroshi has
collected some data and used quadratic regression on his graphing calculator to determine that the relation between angle
2
and distance is given by the equation d = "0.1a + 8.5a " 40 where a is the angle in degrees, and d is the distance in
metres.
a) If Hiroshi kicks the ball at an angle of 60°, how far will the ball go?
d = "0.1(60) 2 + 8.5(60) " 40
! d = "0.1(3600) + 510 " 40
d = "360 + 510 " 40
d = 110
The ball will go 110 m if he kicks it at an angle of 60°.
b) Determine the vertex of the parabola.
!
d = "0.1a 2 + 8.5a " 40
d = "0.1(a 2 " 85a) " 40
d = "0.1(a 2 " 85a + 1806.25 "1806.25) " 40
d = "0.1(a 2 " 85a + 1806.25) + 180.625 " 40
d = "0.1(a " 42.5) 2 + 140.625
The vertex is (42.5, 140.625)
c) Which angle gives the maximum distance?
42.5 degrees
8.
!
A research study has shown that 500 people attend a PHS football game in a tournament when the admission price is $2. In
the championship game, the price will be considered for an increase: for every 20¢ increase, 20 fewer people will attend.
What price will maximize the revenue? What is the value of the maximum revenue?
Let x represent the number of 20¢ increase that we make on the tickets.
Revenue = cost per ticket x number of people
Create the equation
R = (2 + 0.2x)(500 " 20x)
Convert to vertex form
R = 1000 " 40x + 100x " 4 x 2
R = "4(x 2 "15x + 56.25 " 56.25) + 1000
R = "4 x 2 + 60x + 1000
R = "4(x 2 "15x + 56.25) + 225 + 1000
R = "4(x 2 "15x) + 1000
R = "4(x " 7.5) 2 + 1225
If we increase the price 7.5 times, we will maximize the revenue.
2 + 0.2(7.5) = 3.5
The cost of tickets would be $3.50 and the maximum revenue would be $1225.
!
!
9.
A rectangular field is to be enclosed with 600 m of fencing. What dimensions will produce a maximum area?
Let x represent the width. Then the length is 300 – x.
Convert to vertex form
Create the equation
A = x(300 " x)
A = "(x 2 " 300x)
A = "x 2 + 300x
A = "(x 2 " 300x + 22500 " 22500)
A = "(x 2 " 300x + 22500) + 22500
A = "(x "150) 2 + 22500
x = 150 will produce the maximum area. That means the width is 150 m and the length is 150 m.
!
10. A field is bounded
on one side by a river. The!field is to be enclosed on three sides by a fence, to create a rectangular
enclosure. The total length of fence to be used is 100 m. What dimensions will produce a maximum area?
100 - 2x
x
River
Create the equation
A = x(100 " 2x)
Convert to vertex form
A = 100x " 2x 2
A = "2(x 2 " 50x + 625 " 625)
A = "2x 2 + 100x
A = "2(x 2 " 50x + 625) + 1250
A = "2(x 2 " 50x)
A = "2(x " 25) 2 + 1250
The maximum area occurs when x=25.
Therefore, the dimensions of the enclosure of maximum area is 25 m by 50 m.
! having 15 cm as the sum of its !
11. What is the maximum area of a triangle
base and height?
x
15 – x
Convert to vertex form
Create the equation
bh
A=
2
x(15 " x)
A=
2
1 2 15
A=" x + x
2
2
A = "0.5x 2 + 7.5x
A = "0.5(x 2 "15x)
A = "0.5(x 2 "15x + 56.25 " 56.25)
A = "0.5(x 2 "15x + 56.25) + 28.125
A = "0.5(x " 7.5) 2 + 28.125
Therefore the maximum area is 28.125 cm2.
!
!
Bonus
10. A rectangle has perimeter P. Find the maximum possible area of the rectangle.
Let x represent the length. Then the width is
P " 2x
2
Write an equation for the area.
Complete the square to find the vertex.
# P " 2x &
A = x%
(
$ 2 '
#P
&
A = x% " x (
$2
'
P
A = "x 2 + x
2
#
P
A = "% x 2 "
$
2
#
P
A = "%% x 2 "
2
$
#
P
A = "%% x 2 "
2
$
!
&
x(
'
# P &2 # P &2 &
x + % ( " % ( ((
$4' $4' '
#P&
x +% (
$4'
2
#
P & P2
A = "% x " ( +
$
4 ' 16
!
2
P2
The maximum area is
units2.
16
& # P &2
(( + % (
' $4'
!