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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 3 - 2013 - Paper 7
Question 3
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 point where th... 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 3 - 2013 - Paper 7
Step 1
i) y = f(x)
Answer
To sketch the graph of y = f(x) = ln x:
- The domain is x > 0, as the natural logarithm is defined only for positive values.
- The graph increases from negative infinity at x = 0 to positive infinity as x approaches infinity.
- The y-intercept is at (1, 0), where ( f(1) = 0 ).
Asymptote: The graph approaches the vertical asymptote at x = 0. Equation: x = 0.
Step 2
ii) y = |f(x)|
Answer
To sketch the graph of y = |f(x)|:
- For x in (0, 1), f(x) will produce negative values, reflected in the positive region due to the absolute value.
- This means that the graph goes from (0, ∞) to (1, 0) and then increases again into positive y-values as x continues to increase.
- Thus, the graph is V-shaped around the point (1, 0).
Asymptote: The same vertical asymptote at x = 0. Equation: x = 0.
Step 3
iii) y = -f(x - 4)
Answer
To sketch the graph of y = -f(x - 4):
- First, shift the basic ln function horizontally to the right by 4 units, which modifies the x-values.
- The graph now crosses the x-axis at x = 5 (( f(5) = 0 )), resulting in a reflection about the x-axis.
- Thus, it approaches negative infinity as x approaches 4 from the right.
Asymptote: The vertical asymptote at x = 4 due to the shift. Equation: x = 4.