Graph of Functions, Long Questions (Question 2)


Question 2:
(a) The following table shows the corresponding values of x and y for the 
equation y = –x3 + 3x + 1. 

x
–3
–2
–1
0
1
2
3
3.5
4
y
19
3
r
1
3
–1
s
–31.4
–51

Calculate the value of r and s.

(b) For this part of the question, use graph paper. You may use a flexible curve rule.
By using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph of  y = –x3 + 3x + 1 for –3 ≤ x ≤ 4 and –51 ≤ y ≤ 19.

(c) From your graph, find
(iThe value of y when x2.8,
(iiThe value of x when y = 30.

(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation –x3 + 13x – 9 = 0 for –3 ≤ x ≤ 4 and –51 ≤ y ≤ 19.

Solution:
(a)
y = –x3 + 3x + 1 
when x = –1,
r = – (–1)3 + 3(–1) + 1
  = 1 – 3 + 1 = –1
when x = 3,
s = – (3)3 + 3(3) + 1 = –17

(b)

(c) 
(iFrom the graph, when x2.8, y = 15
(iiFrom the graph, when y = 30, x = 3.5

(d)
y = –x3 + 3x + 1 ----- (1)
x3 + 13x – 9 = 0 ----- (2)

y = –x3 + 3x + 1 ----- (1)
0 = –x3 + 13x – 9 ------ (2) ← (Rearrange (2))
(1)  – (2) : y = –10x + 10

The suitable straight line is y = –10x + 10.

Determine the x-coordinates of the two points of intersection of the curve 
y = –x3 + 3x + 1 and the straight line y = –10x + 10.

x
0
4
y = 10x + 10
10
–30


From the graph, x = 0.7, 3.25.