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 $( - 2,{\text{ }}6)$.