The curve \(y=x^3-3x^2\) is drawn on the grid for \(-1\le x\le 4\). (a) Write down the coordinates of the two points where the curve meets the \(x\)-axis. [...

(a) The curve meets the \(x\)-axis where \(y = 0\), so factorise and use the zero-product rule:

This gives \(x = 0\) or \(x = 3\), and both lie inside the drawn range \(-1 \le x \le 4\). Since these are points on the \(x\)-axis, their \(y\)-coordinates are \(0\):

\((0,0)\) [B1] and \((3,0)\) [B1]

Do not divide through by \(x^2\), as that would lose the solution at the origin. At \(x = 0\) the curve touches the axis rather than passing through it, because the repeated factor \(x^2\) keeps the sign of \(y\) the same on both sides.

(b) The solutions of \(x^3 - 3x^2 = -2\) are the \(x\)-coordinates where the curve meets the horizontal line \(y = -2\), so the answer is the number of intersections.

The curve starts at \(y = -4\) when \(x = -1\), rises to \(y = 0\) at the origin, falls to its lowest point \(y = -4\) at \(x = 2\), then climbs steeply to \(y = 16\) at \(x = 4\). The line \(y = -2\) therefore cuts it once on the rising section before the origin, once on the falling section after it, and once on the final rising section, giving

\(3\) solutions [B1]

The point to hold on to is that solving an equation graphically means counting intersections with the appropriate line, not counting where the curve crosses the \(x\)-axis.