(a)     Write down the $x$-intercepts of the graph of $f$ .

(b)     Find the coordinates of the vertex of the graph of $f$ .

Write down the $x$-intercepts of the graph of $f$ .

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

(a)     $x = 1$ , $x = - 3$ (accept ($1$, $0$), ($- 3$, $0$) )     A1A1     N2

[2 marks]

(b)     METHOD 1

attempt to find $x$-coordinate     (M1)

eg   $\frac{{1 + - 3}}{2}$ , $x = \frac{{ - b}}{{2a}}$ , $f'(x) = 0$

correct value, $x = - 1$ (may be seen as a coordinate in the answer)     A1

attempt to find their $y$-coordinate     (M1)

eg   $f( - 1)$ , $- 2 \times 2$ , $y = \frac{{ - D}}{{4a}}$

$y = - 4$     A1

vertex ($- 1$, $- 4$)     N3  

METHOD 2

attempt to complete the square     (M1)

eg   ${x^2} + 2x + 1 - 1 - 3$ 

attempt to put into vertex form     (M1)

eg   ${(x + 1)^2} - 4$ , ${(x - 1)^2} + 4$

vertex ($- 1$, $- 4$)     A1A1     N3

[4 marks]

$x = 1$ , $x = - 3$ (accept ($1$, $0$), ($- 3$, $0$) )     A1A1     N2

[2 marks]

METHOD 1

attempt to find $x$-coordinate     (M1)

eg   $\frac{{1 + - 3}}{2}$ , $x = \frac{{ - b}}{{2a}}$ , $f'(x) = 0$

correct value, $x = - 1$ (may be seen as a coordinate in the answer)     A1

attempt to find their $y$-coordinate     (M1)

eg   $f( - 1)$ , $- 2 \times 2$ , $y = \frac{{ - D}}{{4a}}$

$y = - 4$     A1

vertex ($- 1$, $- 4$)     N3  

METHOD 2

attempt to complete the square     (M1)

eg   ${x^2} + 2x + 1 - 1 - 3$ 

attempt to put into vertex form     (M1)

eg   ${(x + 1)^2} - 4$ , ${(x - 1)^2} + 4$

vertex ($- 1$, $- 4$)     A1A1     N3

[4 marks]

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging $x$-intercepts, using $x = \frac{{ - b}}{{2a}}$ , completing the square.

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging $x$-intercepts, using $x = \frac{{ - b}}{{2a}}$ , completing the square.

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging $x$-intercepts, using $x = \frac{{ - b}}{{2a}}$ , completing the square.