graphs 2002. question one the teachers at valley high want to know the cost of hiring a bus to...
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Graphs
2002
Question One
• The teachers at Valley High want to know the cost of hiring a bus to travel to the interschool sports day.
• Betta Buses has two prices for hiring buses: one for a 45-seat bus and one for a 60-seat bus.
The graph below shows the cost for distances travelled up to 100
kilometres.
Another bus company, Cheap Coaches, gives the teachers an equation to work out the cost of their bus: C = 2d + 200where C is the cost to run the bus for a given distance (in dollars)
and d is the distance travelled by the bus (in kilometres).
Draw the graph for the Cheap Coaches bus on the axes above. You should show costs for distances up to 100 kilometres.
Another bus company, Cheap Coaches, gives the teachers an equation to work out the cost of their bus: C = 2d + 200where C is the cost to run the bus for a given distance (in dollars)
and d is the distance travelled by the bus (in kilometres).
Draw the graph for the Cheap Coaches bus on the axes above. You should show costs for distances up to 100 kilometres.
For what distance is it the same price to hire a 60-seat Betta Buses bus and a Cheap Coaches bus?
For what distance is it the same price to hire a 60-seat Betta Buses bus and a Cheap Coaches bus?
50 km
How is this shown by the graphs?
How is this shown by the graphs?
The intersection point of the lines
Which TWO buses cost the same to hire per kilometre?
Which TWO buses cost the same to hire per kilometre?
The cheap Coaches and the 45 seat bus
How is this shown by the graphs?
The cheap Coaches and the 45 seat bus
How is this shown by the graphs?
The gradients of the lines are the same.
What is the cost per kilometre to hire the 60-seat Betta Buses bus?
What is the cost per kilometre to hire the 60-seat Betta Buses bus?
300/100=$3 per km
How is this shown by the graphs
What is the cost per kilometre to hire the 60-seat Betta Buses bus?
The gradient of the line.
300/100=$3 per km
Use the graph to write the equation for the 45-seat Betta Buses bus. (C = …)
Use the graph to write the equation for the 45-seat Betta Buses bus. (C = …)
C =2d+100
Use the graph to write the equation for the 60-seat Betta Buses bus. (C = …)
Use the graph to write the equation for the 60-seat Betta Buses bus. (C = …)
C =3d+150
Question Two
• Tom is designing a logo for the Valley High sports jacket. Part of the logo is shown on the axes below.
Write the equation of the line labelled AB on the axes below.
−
−
−
−
−
−
−
−
−
− − − − −
The line passes through -2 on the y-axis and gradient is negative 1/2
−
−
−
−
−
−
−
−
−
− − − − −
The line passes through -2 on the y-axis and gradient is negative 1/2
−
−
−
−
−
−
−
−
−
− − − − −
y =−12
x−2
The logo design also has two parabolas.
• On the axes above, draw the graph of y = (x + 4)(x – 1).
On the axes above, draw the graph of y =
(x + 4)(x – 1).
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
On the axes above, draw the graph of y =
(x + 4)(x – 1).
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
On the axes above, draw the graph of y =
(x + 4)(x – 1).
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
21
2⎛⎝⎜
⎞⎠⎟
2
=614
On the axes above, draw the graph of y =
(x + 4)(x – 1).
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
21
2⎛⎝⎜
⎞⎠⎟
2
=614
On the axes above, draw the graph of y =
2x 2 – 8
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
On the axes above, draw the graph of y =
2x 2 – 8
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
The part of the graph that is below the x-axis shows the logo for the sports jacket.
x and y are both measured in centimetres.
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
What is the height of the logo?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
What is the height of the graph?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
8cm
How is this shown by the graphs?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
8cm
How is this shown by the graphs?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
The distance from the vertex to the x-axis
What is the width of the logo?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
What is the width of the logo?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
6 cm
How is this shown by the graphs?
-9-8-7-6-5-4-3-2-1012
-5 -4 -3 -2 -1 0 1 2 3 4 5
The distance between the outer intercepts on the x -axis (-4 and 2)
Question Three
• Maria won the javelin field event on the sports day.• Her coach recorded the heights of the javelin during her
winning throw.• The javelin is 2 metres above the ground when it is released.• During the first 10 seconds of the javelin’s flight, its height
increases at a constant rate. The rate is 3 metres per second.• The remainder of the javelin’s flight can be modelled by a
parabola.• The javelin reaches a maximum height of 36 metres after 12
seconds.
The coach’s data is graphed on the set of axes below.
Write TWO equations to model the flight of Maria’s winning
javelin throw.
The javelin is 2 metres above the ground when it is released.
2
During the first 10 seconds of the javelin’s flight, its height increases at a constant rate. The rate is
3 metres per second.
2
H =3t+ 2
The remainder of the javelin’s flight can be modelled
by a parabola.
2
The javelin reaches a maximum height of 36 metres
after 12 seconds.
2
12
36
The javelin reaches a maximum height of 36 metres
after 12 seconds.
2
12
36
H =−k t−12( )2 + 36
Now find a point on the graph you are certain of.
2
12
36
H =−k t−12( )2 + 36
We know that H = 32 when t = 10
2
12
36
H =−k t−12( )2 + 36
Substitute H = 32 and t = 10
2
12
36
Video cameras were placed 27 metres above the ground.
Calculate the TWO times when the javelin was level with the video cameras.
Show all working.
2
12
36
Substitute 27 for H
2
12
36
H =− t−12( )2 + 36
27 =− t−12( )2 + 36
9 = t−12( )2
t−12 =±3t=15, 9
It wasn’t a parabola at t = 9, so we can only use 15
Substitute 27 for H in the first equation
2
12
36
H =3t+ 227 =3t+ 225 =3t
t=813
It wasn’t a parabola at t = 9, so we can only use 15
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