(a) Copy and complete the following table of values for the relation \(y = 2x^{2} - 7x - 3\). x -2 -1 0 1 2 3 4 5 y 19 -3 -9 (b) Using 2 cm to 1 unit on the...
Question 1 Report
(a) Copy and complete the following table of values for the relation \(y = 2x^{2} - 7x - 3\).
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 19 | -3 | -9 |
(b) Using 2 cm to 1 unit on the x- axis and 2 cm to 5 units on the y- axis, draw the graph of \(y = 2x^{2} - 7x - 3\) for \(-2 \leq x \leq 5\).
(c) From your graph, find the : (i) minimum value of y ;
(ii) gradient of the curve at x = 1.
(d) By drawing a suitable straight line, find the values of x for which \(2x^{2} - 7x - 5 = x + 4\).
Answer Details
(a) For \(y=2x^2-7x-3\), the completed table is:
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|---|---|---|
| \(y\) | \(19\) | \(6\) | \(-3\) | \(-8\) | \(-9\) | \(-6\) | \(1\) | \(12\) |
(b) The points were plotted using the stated scales, and joined with a smooth curve. The straight line \(y=x+6\) and the tangent at \(x=1\) are included for parts (c) and (d).
(c)
(i) The lowest point of the curve gives the minimum value
\[\boxed{y\approx -9}\]
(ii) The tangent at \(x=1\) passes through \((1,-8)\). Using two points on the tangent, for example \((0,-5)\) and \((1,-8)\),
\[\text{gradient}=\frac{-8-(-5)}{1-0}=\boxed{-3}.\]
(d) Rearrange the equation in terms of the curve already drawn:
\[2x^2-7x-5=x+4\]
\[2x^2-7x-3=x+6.\]
Thus, draw the straight line \(y=x+6\). From its points of intersection with the parabola on the graph,
\[\boxed{x\approx -0.9\ \text{ or }\ x\approx 4.9}.\]
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