(a) Copy and complete the table. \(y = x^{2} - 2x - 2\) for \(-4 \leq x \leq 4\) x -4 -3 -2 -1 0 1 2 3 4 y 22 -2 1 6 (b) Using a scale of 2 cm to 1 unit on ...

Answer Details

(a) Completing the table. For each value of \(x\) we evaluate \(y = x^{2} - 2x - 2\) by building it up in rows:

The three missing entries are \(y=13\) (at \(x=-3\)), \(y=1\) (at \(x=-1\)) and \(y=-3\) (at \(x=1\)).

(b) Graph of \(y = x^{2} - 2x - 2\). Plotting the nine points \((x,y)\) and joining them with a smooth curve gives the parabola below. The dashed horizontal line \(y = 2.5\) is used in part (c)(ii) and the vertical dashed line \(x = 1\) is the line of symmetry from part (c)(iii).

(c) Using the graph.

(i) Roots of \(x^{2} - 2x - 2 = 0\). These are the values of \(x\) where the curve cuts the \(x\)-axis, i.e. where \(y = 0\). The curve crosses the \(x\)-axis at \[ x \approx -0.7 \quad \text{and} \quad x \approx 2.7. \]

(ii) Values of \(x\) for which \(x^{2} - 2x - 4\tfrac{1}{2} = 0\). Rearranging so that the left side becomes our plotted expression: \[ x^{2} - 2x - 4\tfrac{1}{2} = 0 \;\Rightarrow\; x^{2} - 2x - 2 = 2\tfrac{1}{2} = 2.5. \] So we draw the horizontal line \(y = 2.5\) and read off where it meets the curve: \[ x \approx -1.3 \quad \text{and} \quad x \approx 3.3. \]

(iii) Equation of the line of symmetry. The lowest point (vertex) of the curve occurs at \(x = \dfrac{-(-2)}{2(1)} = 1\), where \(y = -3\). The curve is symmetrical about the vertical line through this point, so the line of symmetry is \[ x = 1. \]