Tomas grows vegetables on a garden allotment and writes a page for the allotment club newsletter. For the page he draws the graph of \(y = x^2 - 3x - 2\) fo...

Drawing a quadratic graph accurately depends on getting every table value right, since one wrong point distorts the whole curve. The graph then gives estimates for the roots and shows the minimum point.

(a) Complete the table. [2]
Substitute into \(y = x^2 - 3x - 2\), taking care with the signs when \(x\) is negative:

Note \(-3 \times (-1) = +3\); treating it as \(-3\) gives \(y = -4\) and spoils the shape of the curve.

(b) Draw the graph. [2]
Plot the eight points and join them with a single smooth curve, not a series of straight segments:

(c) Estimates for the solutions of \(x^2 - 3x - 2 = 0\). [2]
The solutions are the \(x\) values where the curve crosses the horizontal axis, marked above. Reading off:

The quadratic formula confirms these: \(x = \dfrac{3 \pm \sqrt{9 + 8}}{2} = \dfrac{3 \pm \sqrt{17}}{2}\), giving \(-0.561\ldots\) and \(3.561\ldots\)

(d) Minimum point. [1]
The table shows equal values of \(-4\) at \(x = 1\) and \(x = 2\), so by symmetry the lowest point is halfway between them, at \(x = 1.5\):

The minimum point is \((1.5,\ -4.25)\).

Examination point: when two table values are equal, the turning point lies exactly halfway between those two \(x\) values. That is faster and more accurate than trying to read the lowest point of a hand-drawn curve, and it also gives the line of symmetry, \(x = 1.5\).