19M.1.AHL.TZ2.H_3

pestleMathematicsAIHLPaper 119M· ahl-2-8-transformations-of-graphs-composite-transformationssource ↗

Consider the function $f\left( x \right) = {x^4} - 6{x^2} - 2x + 4$ , $x \in \mathbb{R}$ .

The graph of $f$ is translated two units to the left to form the function $g\left( x \right)$ .

Express $g\left( x \right)$ in the form $a{x^4} + b{x^3} + c{x^2} + dx + e$ where $a$ , $b$ , $c$ , $d$ , $e \in \mathbb{Z}$ .

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$g\left( x \right) = f\left( {x + 2} \right)\left( { = {{\left( {x + 2} \right)}^4} - 6{{\left( {x + 2} \right)}^2} - 2\left( {x + 2} \right) + 4} \right)$ M1

attempt to expand ${{{\left( {x + 2} \right)}^4}}$ M1

${\left( {x + 2} \right)^4} = {x^4} + 4\left( {2{x^3}} \right) + 6\left( {{2^2}{x^2}} \right) + 4\left( {{2^3}x} \right) + {2^4}$ (A1)

$= {x^4} + 8{x^3} + 24{x^2} + 32x + 16$ A1

$g\left( x \right) = {x^4} + 8{x^3} + 24{x^2} + 32x + 16 - 6\left( {{x^2} + 4x + 4} \right) - 2x - 4 + 4$

$= {x^4} + 8{x^3} + 18{x^2} + 6x - 8$ A1

Note: For correct expansion of $f\left( {x - 2} \right) = {x^4} - 8{x^3} + 18{x^2} - 10x$ award max M0M1(A1)A0A1.

[5 marks]

Examiners’ report

[N/A]