Using a scale of 2cm to 1 unit on the x- axis and 1cm to 1 unit on the y- axis, draw on the same axes the graphs of \(y = 3 + 2x - x^{2}; y = 2x - 3\) for \(-3 \leq x \leq 4\). Using your graph:
(i) solve the equation \(6 - x^{2} = 0\);
(ii) find the maximum value of \(3 + 2x - x^{2}\);
(iii) find the range of x for which \(3 + 2x - x^{2} \leq 1\), expressing all your answers correct to one decimal place.
Verified Correct Answer: .
Step-by-Step Mathematical Solution
\(y = 3 + 2x - x^{2}\) and \(y = 2x - 3\)
Table of values for the equation for \(-3 \leq x \leq 4\)
x
-3
-2
-1
0
1
2
3
4
\(y = 3 + 2x - x^{2}\)
-12
-5
0
3
4
3
0
5
\(y = 2x - 3\)
-9
-7
-5
-3
-1
1
3
5
(i) \(6 - x^{2} = 0\)
\(6 - 3 + 2x - x^{2} = 2x - 3\)
\(3 + 2x - x^{2} = 2x - 3\)
\(\therefore y = 2x - 3\)
Read the point where the two equations intersect on the graph.
x = -2.6 and x = 2.5.
(ii) Maximum value of \(3 + 2x - x^{2}\) is at y = 4.
(iii) Range for which \(3 + 2x - x^{2} \leq 1\) is represented by the shaded portion in the graph.
Source: Myschool Past Questions Archive (Record #54615)