Find \((f \circ g)(x)\) .

The vector \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 1}
\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of h .

Find the coordinates of the vertex of the graph of h .

The vector \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 1}
\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of h .

Show that \(h(x) = {x^2} - 8x + 19\) .

The vector \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 1}
\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of h .

The line \(y = 2x - 6\) is a tangent to the graph of h at the point P. Find the x-coordinate of P.

attempt to form composition (in any order)     (M1)

\((f \circ g)(x) = {(x - 1)^2} + 4\)    \(({x^2} - 2x + 5)\)     A1     N2

[2 marks]

METHOD 1

vertex of \(f \circ g\) at (1, 4)     (A1)

evidence of appropriate approach     (M1)

e.g. adding \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 1}
\end{array}} \right)\) to the coordinates of the vertex of \(f \circ g\)

vertex of h at (4, 3)     A1     N3

METHOD 2

attempt to find \(h(x)\)     (M1)

e.g. \({((x - 3) - 1)^2} + 4 - 1\) , \(h(x) = (f \circ g)(x - 3) - 1\)

\(h(x) = {(x - 4)^2} + 3\)     (A1)

vertex of h at (4, 3)     A1     N3

[3 marks]

evidence of appropriate approach     (M1)

e.g. \({(x - 4)^2} + 3\) ,\({(x - 3)^2} - 2(x - 3) + 5 - 1\)

simplifying     A1

e.g. \(h(x) = {x^2} - 8x + 16 + 3\) , \({x^2} - 6x + 9 - 2x + 6 + 4\)

\(h(x) = {x^2} - 8x + 19\)     AG     N0

[2 marks]

METHOD 1

equating functions to find intersection point     (M1)

e.g. \({x^2} - 8x + 19 = 2x - 6\) , \(y = h(x)\)

\({x^2} - 10x + 25 + 0\)     A1

evidence of appropriate approach to solve     (M1)

e.g. factorizing, quadratic formula

appropriate working     A1

e.g. \({(x - 5)^2} = 0\)

\(x = 5\)  \((p = 5)\)     A1     N3

METHOD 2

attempt to find \(h'(x)\)     (M1)

\(h(x) = 2x - 8\)     A1

recognizing that the gradient of the tangent is the derivative     (M1)

e.g. gradient at \(p = 2\)

\(2x - 8 = 2\)  \((2x = 10)\)     A1

\(x = 5\)     A1     N3

[5 marks]

Candidates showed good understanding of finding the composite function in part (a).

There were some who did not seem to understand what the vector translation meant in part (b).

Candidates showed good understanding of manipulating the quadratic in part (c).

There was more than one method to solve for h in part (d), and a pleasing number of candidates were successful in this part of the question.