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. 

$$y= \dfrac 20$$​​

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The denominator is $$0$$ so this is undefined. 

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

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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.​

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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

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.

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. 

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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.