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Page 1: 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

Graphs

2002

Page 2: 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

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.

Page 3: 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

The graph below shows the cost for distances travelled up to 100

kilometres.

Page 4: 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

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.

Page 5: 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

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.

Page 6: 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

For what distance is it the same price to hire a 60-seat Betta Buses bus and a Cheap Coaches bus?

Page 7: 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

For what distance is it the same price to hire a 60-seat Betta Buses bus and a Cheap Coaches bus?

50 km

Page 8: 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

How is this shown by the graphs?

Page 9: 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

How is this shown by the graphs?

The intersection point of the lines

Page 10: 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

Which TWO buses cost the same to hire per kilometre?

Page 11: 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

Which TWO buses cost the same to hire per kilometre?

The cheap Coaches and the 45 seat bus

Page 12: 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

How is this shown by the graphs?

The cheap Coaches and the 45 seat bus

Page 13: 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

How is this shown by the graphs?

The gradients of the lines are the same.

Page 14: 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

What is the cost per kilometre to hire the 60-seat Betta Buses bus?

Page 15: 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

What is the cost per kilometre to hire the 60-seat Betta Buses bus?

300/100=$3 per km

Page 16: 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

How is this shown by the graphs

Page 17: 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

What is the cost per kilometre to hire the 60-seat Betta Buses bus?

The gradient of the line.

300/100=$3 per km

Page 18: 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

Use the graph to write the equation for the 45-seat Betta Buses bus. (C = …)

Page 19: 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

Use the graph to write the equation for the 45-seat Betta Buses bus. (C = …)

C =2d+100

Page 20: 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

Use the graph to write the equation for the 60-seat Betta Buses bus. (C = …)

Page 21: 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

Use the graph to write the equation for the 60-seat Betta Buses bus. (C = …)

C =3d+150

Page 22: 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

Question Two

• Tom is designing a logo for the Valley High sports jacket. Part of the logo is shown on the axes below.

Page 23: 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

Write the equation of the line labelled AB on the axes below.

− − − − −

Page 24: 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

The line passes through -2 on the y-axis and gradient is negative 1/2

− − − − −

Page 25: 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

The line passes through -2 on the y-axis and gradient is negative 1/2

− − − − −

y =−12

x−2

Page 26: 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

The logo design also has two parabolas.

• On the axes above, draw the graph of y = (x + 4)(x – 1).

Page 27: 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

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

Page 28: 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

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

Page 29: 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

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

Page 30: 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

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

Page 31: 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

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

Page 32: 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

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

Page 33: 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

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

Page 34: 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

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

Page 35: 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

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

Page 36: 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

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

Page 37: 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

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

Page 38: 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

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

Page 39: 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

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

Page 40: 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

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)

Page 41: 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

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.

Page 42: 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

Write TWO equations to model the flight of Maria’s winning

javelin throw.

Page 43: 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

The javelin is 2 metres above the ground when it is released.

2

Page 44: 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

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

Page 45: 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

The remainder of the javelin’s flight can be modelled

by a parabola.

2

Page 46: 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

The javelin reaches a maximum height of 36 metres

after 12 seconds.

2

12

36

Page 47: 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

The javelin reaches a maximum height of 36 metres

after 12 seconds.

2

12

36

H =−k t−12( )2 + 36

Page 48: 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

Now find a point on the graph you are certain of.

2

12

36

H =−k t−12( )2 + 36

Page 49: 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

We know that H = 32 when t = 10

2

12

36

H =−k t−12( )2 + 36

Page 50: 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

Substitute H = 32 and t = 10

2

12

36

Page 51: 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

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

Page 52: 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

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

Page 53: 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

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