Given that
$f(x) = \ln x, \ x > 0$
sketch on separate axes the graphs of
(i) $y = f(x)$,
(ii) $y = |f(x)|$,
(iii) $y = -f(x - 4)$ - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 7
Question 4
Given that
$f(x) = \ln x, \ x > 0$
sketch on separate axes the graphs of
(i) $y = f(x)$,
(ii) $y = |f(x)|$,
(iii) $y = -f(x - 4)$.
Show, on each diagram, the p... show full transcript
Worked Solution & Example Answer:Given that
$f(x) = \ln x, \ x > 0$
sketch on separate axes the graphs of
(i) $y = f(x)$,
(ii) $y = |f(x)|$,
(iii) $y = -f(x - 4)$ - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 7
Step 1
(i) $y = f(x)$
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Answer
To sketch the graph of y=f(x)=lnx:
The graph passes through the point (1, 0) because ln(1)=0.
As x approaches 0 from the right, lnx approaches −∞, indicating a vertical asymptote at the line x=0.
The graph is increasing and concave down throughout its domain x>0.
Asymptote: The equation of the asymptote is x=0.
Step 2
(ii) $y = |f(x)|$
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Answer
To sketch the graph of y=∣f(x)∣=∣lnx∣:
For x>1, the graph mirrors the original graph of y=lnx.
For 0<x<1, the graph will reflect the negative part of lnx, turning it positive.
The graph meets the point (1, 0) and has a vertical asymptote at x=0.
Asymptote: The equation of the asymptote is x=0.
Step 3
(iii) $y = -f(x - 4)$
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Answer
To sketch the graph of y=−f(x−4)=−ln(x−4):
The graph shifts to the right by 4 units, changing the vertical asymptote to x=4.
The graph will now decrease from 0 at (5,0), heading towards −∞ as x approaches 4 from the right.
