Question

task 7: soccer ball challenge
your next task takes place at the olympic stadium in turin. here, you are required to score a goal on the famous soccer field from team turin. but, will you be able to score? amazingly you see the ball directly before you, but will it hit the field try?

1. the observations for height of the soccer ball at different times are as follows:

time (s)	height (m)	first differences	second differences
0.5	2.25
1	4
1.5	5.25
2	6
2.5	6.25
3	6
3.5	5.25
4	4
4.5	2.25
5	0

a) determine the first difference and second differences. record your answers in the table above. (5 marks)
b) what do you notice about the first difference values? (1 mark)
_____________________________________
c) what do you notice about your second difference values? (1 mark)
_____________________________________
d) what type of function does this data represent?
(linear/quadratic/neither) ______________ (1 mark)
e) give a reason for you choice. (1 mark)
_____________________________________
page 5 of 6

Explanation:

Step1: Calculate first differences

First difference = Current height - Previous height

• $2.25 - 0 = 2.25$
• $4 - 2.25 = 1.75$
• $5.25 - 4 = 1.25$
• $6 - 5.25 = 0.75$
• $6.25 - 6 = 0.25$
• $6 - 6.25 = -0.25$
• $5.25 - 6 = -0.75$
• $4 - 5.25 = -1.25$
• $2.25 - 4 = -1.75$
• $0 - 2.25 = -2.25$

Step2: Calculate second differences

Second difference = Current first difference - Previous first difference

• $1.75 - 2.25 = -0.5$
• $1.25 - 1.75 = -0.5$
• $0.75 - 1.25 = -0.5$
• $0.25 - 0.75 = -0.5$
• $-0.25 - 0.25 = -0.5$
• $-0.75 - (-0.25) = -0.5$
• $-1.25 - (-0.75) = -0.5$
• $-1.75 - (-1.25) = -0.5$
• $-2.25 - (-1.75) = -0.5$

Answer:

Completed Table:

Time (s)	Height (m)	First Differences	Second Differences
0.5	2.25	2.25	-
1	4	1.75	-0.5
1.5	5.25	1.25	-0.5
2	6	0.75	-0.5
2.5	6.25	0.25	-0.5
3	6	-0.25	-0.5
3.5	5.25	-0.75	-0.5
4	4	-1.25	-0.5
4.5	2.25	-1.75	-0.5
5	0	-2.25	-0.5

b)

The first differences decrease by a constant amount.

c)

The second differences are all constant (-0.5).

d)

Quadratic

e)

A function with constant second differences is quadratic.