18N.1.SL.TZ0.S_8

pestleMathematicsAISLPaper 118N· sl-2-5-modelling-functionssource ↗

Let $f\left( x \right) = {x^2} - 4x - 5$ . The following diagram shows part of the graph of $f$ .

The function can be written in the form $f\left( x \right) = {\left( {x - h} \right)^2} + k$ .

Find the equation of the axis of symmetry of the graph of $f$ .

[2]

Write down the value of $h$ .

[1]

c.i.

Find the value of $k$ .

[3]

c.ii.

The graph of a second function, $g$ , is obtained by a reflection of the graph of $f$ in the $y$ -axis, followed by a translation of $\left( {\begin{array}{*{20}{c}} { - 3} \\ 6 \end{array}} \right)$ .

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

[5]

Markscheme / solution

correct working (A1)

eg $\frac{{ - \left( { - 4} \right)}}{{2\left( 1 \right)}}$ , $\frac{{ - 1 + 5}}{2}$

$x = 2$ (must be an equation with $x =$ ) A1 N2

[2 marks]

$h$ = 2 A1 N1

[1 mark]

c.i.

METHOD 1

valid approach (M1)

eg $f$ (2)

correct substitution (A1)

eg (2)² − 4(2) − 5

$k$ = −9 A1 N2

METHOD 2

valid attempt to complete the square (M1)

eg $x$ ² − 4 $x$ + 4

correct working (A1)

eg ( $x$ ² − 4 $x$ + 4) − 4 − 5, ( $x$ − 2)² − 9

$k$ = −9 A1 N2

[3 marks]

c.ii.

METHOD 1 (working with vertex)

vertex of $f$ is at (2, −9) (A1)

correct horizontal reflection (A1)

eg $x$ = −2, (−2, −9)

valid approach for translation of their $x$ or $y$ value (M1)

eg $x$ − 3, $y$ + 6, $\left( {\begin{array}{*{20}{c}} { - 2} \\ { - 9} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} { - 3} \\ 6 \end{array}} \right)$ , one correct coordinate for vertex

vertex of $g$ is (−5, −3) (accept $x$ = −5, $y$ = −3) A1A1 N1N1

METHOD 2 (working with function)

correct approach for horizontal reflection (A1)

eg $f$ (− $x$ )

correct horizontal reflection (A1)

eg (− $x$ )² −4(− $x$ ) − 5, $x$ ²+ 4 $x$ − 5, (− $x$ − 2)² − 9

valid approach for translation of their $x$ or $y$ value (M1)

eg ( $x$ + 3)² + 4( $x$ + 3) − 5 + 6, $x$ ² + 10 $x$ + 22, ( $x$ + 5)² − 3, one correct coordinate for vertex

vertex of $g$ is (−5, −3) (accept $x$ = −5, $y$ = −3) A1A1 N1N1

[5 marks]

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.

[N/A]