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.

(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.