Find \(f( - 2)\).

Find \(f'(x)\).

Find the gradient of the graph of \(f\) at \(x =  - 2\).

Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\).

Write down the equation of \(T\).

Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\).

Sketch the graph of \(f\) for \( - 5 \leqslant x \leqslant 5\) and \( - 20 \leqslant y \leqslant 20\).

Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\).

Draw \(T\) on your sketch.

The tangent, \(T\), intersects the graph of \(f\) at a second point, P.

Use your graphic display calculator to find the coordinates of P.

\(0.5 \times {( - 2)^2} - \frac{8}{{ - 2}}\)     (M1)

Note: Award (M1) for substitution of \(x =  - 2\) into the formula of the function.

\(6\)     (A1)(G2)

\(f'(x) = x + 8{x^{ - 2}}\)     (A1)(A1)(A1)

Notes: Award (A1) for \(x\), (A1) for \(8\), (A1) for \({x^{ - 2}}\) or \(\frac{1}{{{x^2}}}\) (each term must have correct sign). Award at most (A1)(A1)(A0) if there are additional terms present or further incorrect simplifications are seen.

\(f'( - 2) =  - 2 + 8{( - 2)^{ - 2}}\)     (M1)

Note: Award (M1) for \(x =  - 2\) substituted into their \(f'(x)\) from part (b).

\( = 0\)     (A1)(ft)(G2)

Note: Follow through from their derivative function.

\(y = 6\;\;\;\)OR\(\;\;\;y = 0x + 6\;\;\;\)OR\(\;\;\;y - 6 = 0(x + 2)\)     (A1)(ft)(A1)(ft)(G2)

Notes: Award (A1)(ft) for their gradient from part (c), (A1)(ft) for their answer from part (a). Answer must be an equation.

Award (A0)(A0) for \(x = 6\).

     (A1)(A1)(A1)(A1)

Notes: Award (A1) for labels and some indication of scales in the stated window. The point \((-2,{\text{ }}6)\) correctly labelled, or an \(x\)-value and a \(y\)-value on their axes in approximately the correct position, are acceptable indication of scales.

Award (A1) for correct general shape (curve must be smooth and must not cross the \(y\)-axis).

Award (A1) for \(x\)-intercept in approximately the correct position.

Award (A1) for local minimum in the second quadrant.

Tangent to graph drawn approximately at \(x =  - 2\)     (A1)(ft)(A1)(ft)

Notes: Award (A1)(ft) for straight line tangent to curve at approximately \(x =  - 2\), with approximately correct gradient. Tangent must be straight for the (A1)(ft) to be awarded.

Award (A1)(ft) for (extended) line passing through approximately their \(y\)-intercept from (d). Follow through from their gradient in part (c) and their equation in part (d).

\((4,{\text{ }}6)\;\;\;\)OR\(\;\;\;x = 4,{\text{ }}y = 6\)     (G1)(ft)(G1)(ft)

Notes: Follow through from their tangent from part (d). If brackets are missing then award (G0)(G1)(ft).

If line intersects their graph at more than one point (apart from \(( - 2,{\text{ }}6)\)), follow through from the first point of intersection (to the right of \( - 2\)).

Award (G0)(G0) for \(( - 2,{\text{ }}6)\).