Question 7:
(c) From your graph, find
(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation –x^{3} + 13x – 9 = 0 for –3 ≤ x ≤ 4 and –51 ≤ y ≤ 19.
Solution:
(d)
(a) The following table shows the corresponding values of x and y for the
equation y= –x^{3} + 3x + 1.
equation y= –x^{3} + 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 xaxis and 2cm to 5 units on the yaxis, draw the graph of y = –x^{3} + 3x + 1 for –3 ≤ x ≤ 4 and –51 ≤ y ≤ 19.
(c) From your graph, find
(i) The value of y when x = –2.8,
(ii) The 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 –x^{3} + 13x – 9 = 0 for –3 ≤ x ≤ 4 and –51 ≤ y ≤ 19.
(a)
y= –x^{3} + 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)
(i) From the graph, when x = –2.8, y = 15
(ii) From the graph, when y = –30, x = 3.5
(d)
y= –x^{3} + 3x+ 1 —– (1)
–x^{3}+ 13x – 9 = 0 —– (2)
y= –x^{3} + 3x + 1 —– (1)
0 = –x^{3} + 13x – 9 —— (2) ← (Rearrange (2))
(1) – (2) : y = –10x + 10
The suitable straight line is y = –10x + 10.
Determine the xcoordinates of the two points of intersection of the curve
y = –x^{3} + 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.
Question 8:
(a) Complete the table in the answer space for the equation y = x^{3} – 4x – 10 by writing down the values of y when x = –1 and x = 3.
(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 xaxis and 2cm to 10 units on the yaxis, draw the graph of y = x^{3} – 4x – 10 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.
(c) From your graph, find
(i) the value of y when x = 2.2,
(ii) the value of x when y = 15.
(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation x^{3} – 12x – 5 = 0 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.
Answer:
Solution:
(a)
y = x^{3} – 4x – 10
when x = –1,
y = (–1)^{3} – 4(–1) – 10
= –7
when x = 3,
y = (3)^{3} – 4(3) – 10
= 5
(b)
(c)
(i) From the graph, when x = 2.2, y = –8
(ii) From the graph, when y = 15, x = 3.4
(d)
y = x^{3} – 4x – 10  (1)
0 = x^{3} – 12x – 5  (2)
(1) – (2) : y = 8x – 5
The suitable straight line is y = 8x–5. Determine the xcoordinates of the two points of intersection of the curve y = x^{3} – 4x – 10 and the straight line y = 8x –5.
From the graph, x = –0.45, 3.7.
(a) Complete the table in the answer space for the equation y = x^{3} – 4x – 10 by writing down the values of y when x = –1 and x = 3.
(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 xaxis and 2cm to 10 units on the yaxis, draw the graph of y = x^{3} – 4x – 10 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.
(c) From your graph, find
(i) the value of y when x = 2.2,
(ii) the value of x when y = 15.
(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation x^{3} – 12x – 5 = 0 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.
Answer:
x 
–3 
–2 
–1 
0 
1 
2 
3 
3.5 
4 
y 
–25 
–10 
–10 
–13 
–10 
18.9 
38 
(a)
y = x^{3} – 4x – 10
when x = –1,
y = (–1)^{3} – 4(–1) – 10
= –7
when x = 3,
y = (3)^{3} – 4(3) – 10
= 5
(b)
(c)
(i) From the graph, when x = 2.2, y = –8
(ii) From the graph, when y = 15, x = 3.4
(d)
y = x^{3} – 4x – 10  (1)
0 = x^{3} – 12x – 5  (2)
(1) – (2) : y = 8x – 5
The suitable straight line is y = 8x–5. Determine the xcoordinates of the two points of intersection of the curve y = x^{3} – 4x – 10 and the straight line y = 8x –5.
x 
0 
2 
y = 8x – 5 
–5 
–11
