Figure 1 shows a sketch of the curve with equation $y = f(x)$ where $$f(x) = \frac{x}{x - 2},\quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations $y = 1$ and $x = 2$, as shown in Figure 1. (a) In the space below, sketch the curve with equation $y = f(x - 1)$ and state the equations of the asymptotes of this curve. (b) Find the coordinates of the points where the curve with equation $y = f(x - 1)$ crosses the coordinate axes.

A-Level Edexcel Maths Pure Functions: Figure 1 shows a sketch of the c

Figure 1 shows a sketch of the curve with equation $y = f(x)$ where $$f(x) = \frac{x}{x - 2},\quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations $y = 1$ and $x = 2$, as shown in Figure 1 - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2

Question 7

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Figure 1 shows a sketch of the curve with equation $y = f(x)$ where $$f(x) = \frac{x}{x - 2},\quad x \neq 2$$ The curve passes through the origin and has two asym... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve with equation $y = f(x)$ where $$f(x) = \frac{x}{x - 2},\quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations $y = 1$ and $x = 2$, as shown in Figure 1 - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2

Step 1

Sketch the curve with equation $y = f(x - 1)$ and state the equations of the asymptotes of this curve.

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Answer

To find the new curve $y = f(x - 1)$ , substitute $(x - 1)$ into the function:

$f(x - 1) = \frac{x - 1}{(x - 1) - 2} = \frac{x - 1}{x - 3}$

Asymptotes:

The vertical asymptote occurs when the denominator is zero: $x - 3 = 0 \Rightarrow x = 3$
The horizontal asymptote is determined by the behavior as $x \to \infty$ : $y = 1$

This leads us to the following equations of asymptotes:

$y = 1$ (horizontal asymptote)
$x = 3$ (vertical asymptote)

The sketch should show the curve approaching these asymptotes.

Step 2

Find the coordinates of the points where the curve with equation $y = f(x - 1)$ crosses the coordinate axes.

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Answer

To find the x-intercept, set $y = 0$ :

$0 = \frac{x - 1}{x - 3} \Rightarrow x - 1 = 0 \Rightarrow x = 1$

Thus, the x-intercept is $(1, 0)$ .

To find the y-intercept, set $x = 0$ :

$y = \frac{0 - 1}{0 - 3} = \frac{-1}{-3} = \frac{1}{3}$

Thus, the y-intercept is $(0, \frac{1}{3})$ .

Coordinates of intercepts:

x-intercept: $(1, 0)$
y-intercept: $(0, \frac{1}{3})$

Figure 1 shows a sketch of the curve with equation $y = f(x)$ where $$f(x) = \frac{x}{x - 2},\quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations $y = 1$ and $x = 2$, as shown in Figure 1 - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2

Sketch the curve with equation $y = f(x - 1)$ and state the equations of the asymptotes of this curve.

Find the coordinates of the points where the curve with equation $y = f(x - 1)$ crosses the coordinate axes.

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