(a) Copy and complete the table for \(y = 3x^{2} - 5x - 7\) x -3 -2 -1 0 1 2 3 4 \(y = 3x^{2} - 5x - 7\) 35 -7 -9 5 (b) Using a scale of 2cm = 1 unit along ...
Question 1 Report
(a) Copy and complete the table for \(y = 3x^{2} - 5x - 7\)
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| \(y = 3x^{2} - 5x - 7\) | 35 | -7 | -9 | 5 |
(b) Using a scale of 2cm = 1 unit along the x- axis and 2cm = 5 units along the y- axis, draw the graph of \(y = 3x^{2} - 5x - 7\).
(c) On the same axis, draw the graph of \(y + 3x + 2 = 0\).
(d) From your graph, find the : (i) range of values of x for which \(3x^{2} - 5x - 7 < 0\) ; (ii) roots of the equation \(3x^{2} - 2x - 5 = 0\).
Answer Details
(a) For \(y=3x^{2}-5x-7\), the completed table is:
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|---|---|
| \(y=3x^2-5x-7\) | 35 | 15 | 1 | −7 | −9 | −5 | 5 | 21 |
For example, when \(x=-2\),
\[ y=3(-2)^2-5(-2)-7=12+10-7=15. \](b) and (c) The required graphs, drawn on the same axes, are shown below. The curve is \(y=3x^2-5x-7\), and the straight line is \(y=-3x-2\).
(d)(i) \(3x^2-5x-7<0\) where the parabola lies below the \(x\)-axis. Reading the intercepts from the graph gives approximately \(-0.9\) and \(2.6\). Hence,
\[ -0.9<x<2.6. \](d)(ii) The roots of \(3x^2-2x-5=0\) are the \(x\)-coordinates of the points where the line and curve intersect:
\[ 3x^2-5x-7=-3x-2 \] \[ 3x^2-2x-5=0. \] From the graph, \[ x=-1.0\quad\text{or}\quad x\approx1.7. \]Indeed, \(3x^2-2x-5=(3x-5)(x+1)\), so the exact roots are \(x=-1\) and \(x=\frac53\).
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