Write down the value of $h$ and of $k$.

Find the value of $c$.

Let $g(x) =  - {(x - 3)^2} + 1$. The graph of $g$ is obtained by a reflection of the graph of $f$ in the $x$-axis, followed by a translation of $\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)$.

Find the value of $p$ and of $q$.

Find the x-coordinates of the points of intersection of the graphs of $f$ and $g$.

$h = 1,{\text{ }}k =  - 9\;\;\;\left( {{\text{accept }}{{(x - 1)}^2} - 9} \right)$     A1A1     N2

[2 marks]

METHOD 1

attempt to substitute $x = 0$ into their quadratic function     (M1)

eg$\;\;\;f(0),{\text{ }}{(0 - 1)^2} - 9$

$c =  - 8$     A1     N2

METHOD 2

attempt to expand their quadratic function     (M1)

eg$\;\;\;{x^2} - 2x + 1 - 9,{\text{ }}{x^2} - 2x - 8$

$c =  - 8$     A1     N2

[2 marks]

evidence of correct reflection     A1

eg$\;\;\; - \left( {{{(x - 1)}^2} - 9} \right)$, vertex at $(1,{\text{ }}9)$, y-intercept at $(0,{\text{ }}8)$

valid attempt to find horizontal shift     (M1)

eg$\;\;\;1 + p = 3,{\text{ }}1 \to 3$

$p = 2$     A1     N2

valid attempt to find vertical shift     (M1)

eg$\;\;\;9 + q = 1,{\text{ }}9 \to 1,{\text{ }} - 9 + q = 1$

$q =  - 8$     A1     N2

Notes:     An error in finding the reflection may still allow the correct values of $p$ and $q$ to be found, as the error may not affect subsequent working. In this case, award A0 for the reflection, M1A1 for $p = 2$, and M1A1 for $q =  - 8$.

If no working shown, award N0 for $q = 10$.

[5 marks]

valid approach (check FT from (a))     M1

eg$\;\;\;f(x) = g(x),{\text{ }}{(x - 1)^2} - 9 =  - {(x - 3)^2} + 1$

correct expansion of both binomials     (A1)

eg$\;\;\;{x^2} - 2x + 1,{\text{ }}{x^2} - 6x + 9$

correct working     (A1)

eg$\;\;\;{x^2} - 2x - 8 =  - {x^2} + 6x - 8$

correct equation     (A1)

eg$\;\;\;2{x^2} - 8x = 0,{\text{ }}2{x^2} = 8x$

correct working     (A1)

eg$\;\;\;2x(x - 4) = 0$

$x = 0,{\text{ }}x = 4$     A1A1     N3

[7 marks]

Total [16 marks]