1. Graphs
NCEA Excellence

2. 2007

3. Question 5c Jake and Ryan are playing volleyball.
Jake is 4.0 metres from the net on one side and Ryan is 1.5 metres from the net on the other
side, as shown in the diagram.
When the ball is hit from one player to the other, the path of the ball can be modelled by a parabola.

5. Question 5C
The height of the ball when it leaves Jake is 1 metre above the ground.
The ball reaches its maximum height of 3 metres above the ground when it is directly above the net.
When Ryan jumps, he can reach to a height of 2.2 metres.
Form an equation to model the path of the ball and use it to determine whether or not Ryan could reach it when he jumps.

6. X = -4, y = 1

7. X = 1.5, y = ?
He will not be able to reach the ball.

8. Question 5a John and Richard were playing with a soccer ball.
The graph shows the height of the ball above the ground, during one kick from John, J, towards
Richard, R.
The height of the ball above the ground is y metres.
The horizontal distance of the ball from John is x metres.
The graph has the equation y = 0.1(9 – x)(x + 1).

9. y = 0.1(9 – x)(x + 1)

10. y = 0.1(9 – x)(x + 1)
Write down the value of the y-intercept and explain what it means in this situation.

11. y = 0.1(9 – x)(x + 1)
Write down the value of the y-intercept and explain what it means in this situation.
X = 0, y = 0.9
This is the initial height of the ball when it is kicked by John.

12. y = 0.1(9 – x)(x + 1)
What is the greatest height of the ball above the ground?

13. y = 0.1(9 – x)(x + 1)
What is the greatest height of the ball above the ground?
When x = 4
Y = 2.5 m 4 9 -1

14. Draw the graph
y = –x(x – 1) – 2
= –x2 + x – 2

15. y = –x(x – 1) – 2 = –x2 + x – 2
You can either substitute in values or draw the graph of
y = –x(x – 1) moved down 2

16. Draw the graph
y = 2x2 – 8

17. y = 2x2 – 8

18. Draw the graph
2y + 3x = 6

19. 2y + 3x = 6

20. Write the equations

22. 2006

23. Question 5 Part of the dirt cycle track is shown on the graph below.
The first section, AB, is a straight line and the second section, BCD, is part of a parabola.

25. The equation of the second section, BCD, is
It begins at B, a horizontal distance of 40 metres from the start.
How high is the start of this second section, B, above ground-level?

26. X = 40, y = 4

27. The point A is 6 metres above the ground
The point A is 6 metres above the ground. What is the gradient of the first section AB?

28. How high above ground-level is the lowest point, C?

29. In the graph below, the dotted line DEF shows the path of a rider performing a jump.
The rider leaves the track at D, jumps to maximum height at E, and lands on a platform at F.
This jump can also be modelled by a parabola.
E is 8 metres above the ground and a horizontal distance of 60 metres from the start.
B and D are both the same height above the ground.
F is a horizontal distance of 72 metres from the start.
Calculate the height of the platform at F.

30. X = 50, y = 4

31. X = 72, y = ?

32. Draw the graph
y = x2 − 6x

33. y = x2 − 6x = x(x - 6)

34. Draw the graph
y = 3 − 2x − x2
= (1 − x)(3 + x)

35. y = 3 − 2x − x2 = (1 − x)(3 + x)

36. Draw the graph
4y − 2x = 8

37. 4y − 2x = 8

38. Write the equations

39. 2005

40. Question 5 Mere buys a small tent for her little sister for Christmas.
The top part is modelled by a parabola.
There are three zips: BE, DE, and EF.
The equation of the frame of the top part of the tent is
where y is the height in cm and x is the distance from the centre line in cm.

41. What is the height of the top of the tent?

42. What is the height of the top of the tent?
80 cm

43. What is the length of the horizontal zip, EF?

44. What is the length of the horizontal zip, EF?
Y = 0
X = 40cm

45. x = 16 X = 67.2 cm The length of AC is 32 cm.
Find the length of zip BE.
x = 16
X = 67.2 cm

46. Two poles, PR and QS help keep the tent upright.
They are 25 cm from the centre-line of the tent.
The bottom part of the tent is also modelled by a parabola.
It cuts the vertical axis at y = –20.
The length FG = 10 cm.
Find the length of the pole PR.

51. 2004

53. How far away from Tere does the ball land?

54. What is the maximum height the ball reaches?
You should find this as accurately as you can.

55. What is the height of the ball when it crosses the half-way line?

56. The Touch team is still warming up before the game.
Lea is 11 metres from the half-way line.
Lea passes the ball towards Jim and it travels as shown in the graph below.

57. Jim is standing 7 metres from the half-way line on the opposite side to Lea. He can reach up a maximum height of 2.2 metres. Will he be able to touch the ball as it goes over his head?