The curve shown in Figure 1 represents the 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 $y = f(x)$ 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 Integration: The curve shown in Figure 1 repr

The curve shown in Figure 1 represents the 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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The curve shown in Figure 1 represents the equation $y = f(x)$, where $$f(x) = \frac{x}{x - 2}, \quad x \neq 2$$ The curve passes through the origin and has two as... show full transcript

Worked Solution & Example Answer:The curve shown in Figure 1 represents the 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 $y = f(x)$ and state the equations of the asymptotes of this curve.

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Answer

To sketch the curve, we observe the function $f(x) = \frac{x}{x - 2}$ . The curve will have vertical and horizontal asymptotes due to the nature of rational functions.

Vertical Asymptote: As $x$ approaches $2$ , the function tends to infinity, indicating a vertical asymptote at: $x = 2$
Horizontal Asymptote: As $x$ approaches infinity or negative infinity, the leading coefficients dominate, and: $y = 1$
Curve Sketch: The curve intersects the y-axis at $(0, 0)$ , and the behavior indicates a single crossing of each axis. The drawn curve has a correct shape with the identified asymptotes clearly marked.

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 new function, we substitute $(x - 1)$ into the original function:

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

Finding intercepts:
- x-intercept: Set $y = 0$ : $\frac{x - 1}{x - 3} = 0 \implies x - 1 = 0 \implies x = 1$
- y-intercept: Set $x = 0$ : $y = \frac{0 - 1}{0 - 3} = \frac{-1}{-3} = \frac{1}{3}$
Coordinates: Thus, the coordinates where the curve crosses the axes are:
- x-axis: $(1, 0)$
- y-axis: $(0, \frac{1}{3})$

The curve shown in Figure 1 represents the 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 $y = f(x)$ 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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