A-Level Edexcel Maths Pure Quadratics: Given that $f(x) = \frac{1}{x}

Given that $f(x) = \frac{1}{x}, \quad x \neq 0$, (a) sketch the graph of $y = f(x) + 3$ and state the equations of the asymptotes - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Question 5

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Given that $f(x) = \frac{1}{x}, \quad x \neq 0$, (a) sketch the graph of $y = f(x) + 3$ and state the equations of the asymptotes. (b) Find the coordinates of... show full transcript

Worked Solution & Example Answer:Given that $f(x) = \frac{1}{x}, \quad x \neq 0$, (a) sketch the graph of $y = f(x) + 3$ and state the equations of the asymptotes - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Step 1

(a) sketch the graph of y = f(x) + 3 and state the equations of the asymptotes.

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Answer

To sketch the graph of $y = f(x) + 3$ , we first recognize that the base function $f(x) = \frac{1}{x}$ has vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$ .

With the transformation applied, the graph shifts upward by 3 units. The new horizontal asymptote is now given by the equation:

$y = 3$

The vertical asymptote remains unchanged at:

$x = 0$

The overall structure of the graph retains two branches, one in the first quadrant and one in the fourth quadrant, without overlapping the asymptotes.

Step 2

(b) Find the coordinates of the point where y = f(x) + 3 crosses a coordinate axis.

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Answer

To find where the graph crosses the coordinate axes, we need to determine where:

The graph crosses the y-axis:
- Set $x = 0$ $x = 0$ :
  - The graph does not cross the y-axis due to the vertical asymptote.
The graph crosses the x-axis:
- Set $y = 0$ :
$0 = \frac{1}{x} + 3$
- Rearranging gives:
  $-3 = \frac{1}{x}$
- Thus,
  $x = -\frac{1}{3}$
- Therefore, the coordinates where $y = f(x) + 3$ crosses the x-axis are:
  $(-\frac{1}{3}, 0)$ .

Given that $f(x) = \frac{1}{x}, \quad x \neq 0$, (a) sketch the graph of $y = f(x) + 3$ and state the equations of the asymptotes - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Worked Solution & Example Answer:Given that $f(x) = \frac{1}{x}, \quad x \neq 0$, (a) sketch the graph of $y = f(x) + 3$ and state the equations of the asymptotes - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

(a) sketch the graph of y = f(x) + 3 and state the equations of the asymptotes.

(b) Find the coordinates of the point where y = f(x) + 3 crosses a coordinate axis.

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