Reciprocal graphs

In a nutshell

Reciprocal functions are inverse function in the format $f(x) = \dfrac {k}{x}$ and $f(x) = \dfrac {k}{x^2}$ where $k$ is a real constant. Reciprocal functions are mainly characterised by asymptotes.

Asymptotes

Asymptotes are lines which a curve approaches but never reaches. Reciprocal functions in the form $\dfrac kx$ and $\dfrac k{x^2}$ have the asymptotes $x=0$ and $y=0$ .

Finding asymptotes

Reciprocal functions have asymptotes because fractions with a denominator of $0$ are undefined.

Example 1

Show that $x=0$ and $y=0$ are asymptotes of $y = \dfrac 2x$ .

Substitute $x=0$ into the equation.

$y= \dfrac 20$

The denominator is $0$ so this is undefined.

$\underline{x=0}$ is an asymptote.

Substitute $y=0$ into the equation.

$\begin{aligned} y&= \dfrac 2{x} \\ \\ x &= \dfrac 2{y} \\ \\ x&= \dfrac20 &\end{aligned}$

The denominator is $0$ , therefore, this is undefined.

$\underline{y=0}$ is an asymptote.

Note: If the $x$ or $y$ axis is an asymptote, there will be no $x$ and $y$ intercepts respectively.

Sketching reciprocal graphs

To sketch a reciprocal graph, there are three important features that need to be identified from a given reciprocal function, $f(x)$ : the asymptotes, the degree of the polynomial and the value of the constant $k$ .

$\boxed{y=\dfrac{k}{x} \ \ and \ \ y=\dfrac{k}{x^2} }$

Procedure

$1.$	Identify the asymptotes of the reciprocal function and mark these lines on the graph.
$2.$	Find the degree of the polynomial which will either be $x^{-1}$ or ${x^{-2}}$ .
$3.$	Discern whether $k$ is positive or negative.
$4.$	For $x^{-1}$ , if $k$ is positive, the curve will be in the $1st$ and $3rd$ quadrant. If $k$ is negative, the curve will be in the $2nd$ and $4th$ quadrant.
$5.$	For $x^{-2}$ , if $k$ is positive, the curve will be in the $1st$ and $2nd$ quadrant. If $k$ is negative, the curve will be in the $3rd$ and $4th$ quadrant.

Note: $x^{-1}=\dfrac 1x$ and $x^{-2}=\dfrac 1{x^2}$ .

Example 2

Sketch the curve $y= \dfrac {-3}{x^2}$ .

Identify the asymptotes.

$y= 0 \$ and $x=0$

The degree of the polynomial is $x^{-2}$ and the constant is negative. This means the curve will be in the $2nd$ and $3rd$ quadrant.

Maths; Sketching graphs; KS5 Year 12; Reciprocal graphs

Note: The dashed lines demarcate the axes. They are used for illustration; draw the axes as normal when sketching graphs.

Comparing reciprocal functions

Reciprocal functions can be sketched differently relative to each other based on the magnitude of their constant value.

Example 3

Sketch the curves $y= \dfrac 2x$ and $y = \dfrac{10}x$ on the same diagram.

Both functions have the asymptotes $y=0$ and $x=0$ . The degree of their polynomials are $x^{-1}$ and the constant values are positive. The graphs will be in the $1st$ and $3rd$ quadrant.

Compare the constant values.

$10>2$

For the same positive values of $x$ :

$y=\dfrac{10}{x} > y= \dfrac 2x$

For the same negative values of $x$ :

$y=\dfrac{10}{x} < y= \dfrac 2x$

Use this information to sketch the curve.

Maths; Sketching graphs; KS5 Year 12; Reciprocal graphs