21M.1.AHL.TZ1.17

pestleMathematicsAIHLPaper 121M· ahl-2-8-transformations-of-graphs-composite-transformationssource ↗

The graph of the function $f (x) = \ln x$ is translated by $(\begin{matrix} a \\ b \end{matrix})$ so that it then passes through the points $(0, 1)$ and $(e^{3}, 1 + \ln 2)$ .

Find the value of $a$ and the value of $b$ .

Markscheme / solution

new function is $f (x - a) + b (= \ln (x - a) + b)$ (M1)

$f (0) = \ln (- a) + b = 1$ A1

$f (e^{3}) = \ln (e^{3} - a) + b = 1 + \ln 2$ A1

$\ln (- a) = \ln (e^{3} - a) - \ln 2$ (M1)

$\ln (- a) = \ln (\frac{e^{3} - a}{2})$

$- a = \frac{e^{3} - a}{2}$

$- 2 a = e^{3} - a$

$a = - e^{3} (= - 20.0855 \dots)$ A1

$b = 1 - \ln e^{3} = 1 - 3 = - 2$ (M1)A1

[7 marks]

Examiners’ report

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