The table below is for the relation \(y = 2 + x - x^{2}\) x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 y -4 -1.75 0 1.25 2 2.25 2 1.25 0 -1.75 -4 (a) Using a scale...
Question 1 Report
The table below is for the relation \(y = 2 + x - x^{2}\)
| x | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| y | -4 | -1.75 | 0 | 1.25 | 2 | 2.25 | 2 | 1.25 | 0 | -1.75 | -4 |
(a) Using a scale of 2cm to 1 unit on each axis, draw the graph of the relation in the interval \(-2 \leq x \leq 3\).
(b) From your graph, find the greatest value of y and the value of x for which this occurs.
(c) Using the same scale and axes, draw the graph of \(y = 1 - x\)
(d) Use your graphs to solve the equation \(1 + 2x - x^{2} = 0\)
Answer Details
(a) Using the scale 2 cm to 1 unit on both axes, plot the given points and draw a smooth curve through them. The completed graph is shown below.
| \(x\) | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y=2+x-x^2\) | -4 | -1.75 | 0 | 1.25 | 2 | 2.25 | 2 | 1.25 | 0 | -1.75 | -4 |
(b) The greatest value of \(y\) is \(2.25\). It occurs when \(x=0.5\).
(c) For \(y=1-x\), use, for example, the points \((-2,3)\), \((0,1)\), \((1,0)\), \((2,-1)\) and \((3,-2)\). Join them with a straight line on the same axes, as shown.
(d) The required equation may be written as
\[1+2x-x^2=0\iff 2+x-x^2=1-x.\]
Hence its solutions are the \(x\)-coordinates of the intersections of \(y=2+x-x^2\) and \(y=1-x\). Reading from the graph gives
\[\boxed{x\approx -0.4\quad\text{or}\quad x\approx 2.4}.\]
(The exact values are \(x=1\pm\sqrt2\).)
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