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

Given that $f(x) = \frac{1}{x}, \; x \neq 0,$ (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

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

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

Worked Solution & Example Answer:Given that $f(x) = \frac{1}{x}, \; x \neq 0,$ (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.

96%

114 rated

Answer

The graph of $y = f(x) + 3$ is a transformation of the basic hyperbola $f(x) = \frac{1}{x}$ , shifted upwards by 3 units.

Asymptotes:

Vertical Asymptote: The vertical asymptote remains at the line $x = 0$ .
Horizontal Asymptote: The horizontal asymptote shifts to the line $y = 3$ .

This results in the graph having branches in the first and fourth quadrants, approaching both asymptotes but never crossing them.

Step 2

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

99%

104 rated

Answer

To find where the graph crosses the y-axis, we set $x = 0$ . However, since $f(x)$ is undefined at $x = 0$ , we cannot find a crossing on the y-axis.

To find where it crosses the x-axis:
Set $y = 0$ :
$0 = \frac{1}{x} + 3$
This simplifies to:
$\frac{1}{x} = -3$
$x = -\frac{1}{3}$
Thus, the coordinates where the graph crosses the x-axis are $(-\frac{1}{3}, 0)$ .

Given that $f(x) = \frac{1}{x}, \; x \neq 0,$ (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 $f(x) = \frac{1}{x}, \; x \neq 0,$ (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.

Asymptotes:

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

Join the A-Level students using SimpleStudy...