Write down the value of the discriminant.

Hence, show that $p = 3$.

The graph of $f$has its vertex on the $x$-axis.

Find the coordinates of the vertex of the graph of $f$.

The graph of $f$ has its vertex on the $x$-axis.

Write down the solution of $f(x) = 0$.

The graph of $f$ has its vertex on the $x$-axis.

The function can be written in the form $f(x) = a{(x - h)^2} + k$. Write down the value of $a$.

The graph of $f$ has its vertex on the $x$-axis.

The function can be written in the form $f(x) = a{(x - h)^2} + k$. Write down the value of $h$.

The graph of $f$ has its vertex on the $x$-axis.

The function can be written in the form $f(x) = a{(x - h)^2} + k$. Write down the value of $k$.

The graph of $f$ has its vertex on the $x$-axis.

The graph of a function $g$ is obtained from the graph of $f$ by a reflection of $f$ in the $x$-axis, followed by a translation by the vector $\left( \begin{array}{c}0\\6\end{array} \right)$. Find $g$, giving your answer in the form $g(x) = A{x^2} + Bx + C$.

correct value $0$, or $36 - 12p$     A2     N2

[2 marks]

correct equation which clearly leads to $p = 3$     A1

eg     $36 - 12p = 0,{\text{ }}36 = 12p$

$p = 3$     AG     N0

[1 mark]

METHOD 1

valid approach     (M1)

eg     $x =  - \frac{b}{{2a}}$

correct working     A1

eg     $- \frac{{( - 6)}}{{2(3)}},{\text{ }}x = \frac{6}{6}$

correct answers     A1A1     N2

eg     $x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)$

METHOD 2

valid approach     (M1)

eg     $f(x) = 0$, factorisation, completing the square

correct working     A1

eg     ${x^2} - 2x + 1 = 0,{\text{ }}(3x - 3)(x - 1),{\text{ }}f(x) = 3{(x - 1)^2}$

correct answers     A1A1     N2

eg     $x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)$

METHOD 3

valid approach using derivative     (M1)

eg     $f'(x) = 0,{\text{ }}6x - 6$

correct equation     A1

eg     $6x - 6 = 0$

correct answers     A1A1     N2

eg     $x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)$

[4 marks]

$x = 1$     A1     N1

[1 mark]

$a = 3$     A1     N1

[1 mark]

$h = 1$     A1     N1

[1 mark]

$k = 0$     A1     N1

[1 mark]

attempt to apply vertical reflection     (M1)

eg     $- f(x),{\text{ }} - 3{(x - 1)^2}$, sketch

attempt to apply vertical shift 6 units up     (M1)

eg     $- f(x) + 6$, vertex $(1, 6)$

transformations performed correctly (in correct order)     (A1)

eg     $- 3{(x - 1)^2} + 6,{\text{ }} - 3{x^2} + 6x - 3 + 6$

$g(x) =  - 3{x^2} + 6x + 3$     A1     N3

[4 marks]