(a) Draw the table of values for the relation \(y = x^{2}\) for the interval \(-3 \leq x \leq 4\). (b) Using a scale of 2 cm to 1 unit on the x- axis and 2 ...
Question 1 Report
(a) Draw the table of values for the relation \(y = x^{2}\) for the interval \(-3 \leq x \leq 4\).
(b) Using a scale of 2 cm to 1 unit on the x- axis and 2 cm to 2 units on the y- axis, draw the graphs of : (i) \(y = x^{2}\) ; (ii) \(y = 2x + 3\) for \(-3 \leq x \leq 4\).
(c) Use your graph to find : (i) the roots of the equation \(x^{2} = 2x + 3\) ; (ii) the gradient of \(y = x^{2}\) at x = -2.
Answer Details
(a) Table of values for \(y=x^2\)
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|---|---|
| \(y=x^2\) | 9 | 4 | 1 | 0 | 1 | 4 | 9 | 16 |
(b) Values for the straight line \(y=2x+3\)
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|---|---|
| \(y=2x+3\) | \(-3\) | \(-1\) | 1 | 3 | 5 | 7 | 9 | 11 |
Using the stated scales, plot the points and draw smooth curve \(y=x^2\) and the straight line \(y=2x+3\). A tangent to the curve at \((-2,4)\) is also shown for part (c)(ii).
(c)(i) The graphs intersect at \((-1,1)\) and \((3,9)\). Hence, the roots of \(x^2=2x+3\) are
\[\boxed{x=-1\text{ and }x=3}.\]
(c)(ii) From the tangent at \(x=-2\), take the points \((-3,8)\) and \((-1,0)\):
\[\text{gradient}=\frac{0-8}{-1-(-3)}=\frac{-8}{2}=\boxed{-4}.\]
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