Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote.
Any graph of a rational function can be obtained from the reciprocal function $f(x)=\dfrac{1}{x}$ by a combination of transformations, including a translation, stretches and compressions.
Type 1: Vertical Compression
\( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{0.5}{x}} \) is compressed vertically from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{0.5}{x}} \lt \color{blue}{y = \frac{1}{x}} \)
Type 2: Vertical Stretch
\( \displaystyle y = \frac{a}{x}, \ a \gt 1 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{2}{x}} \) is stretched vertically from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{2}{x}} \gt \color{blue}{y = \frac{1}{x}} \)
Type 3: Horizontal Stretch
\( \displaystyle y = \frac{1}{bx}, \ 0 \lt b \lt 1 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{0.5x}} \) is stretched horizontally from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{0.5x} = \frac{2}{x}} \gt \color{blue}{y = \frac{1}{x}} \)
Type 4: Horizontal Compression
\( \displaystyle y = \frac{1}{bx}, \ b \gt 1 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{2x}} \) is compressed horizontally from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{2x}} \lt \color{blue}{y = \frac{1}{x}} \)
Type 5: Horizontal Translation to the Right
\( \displaystyle y = \frac{1}{x-c}, \ c \gt 0 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x-1}} \) is translated by \( 1 \) unit to the right from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
Type 6: Horizontal Translation to the Left
\( \displaystyle y = \frac{1}{x+c}, \ c \gt 0 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x+1}} \) is translated by \( 1 \) unit to the left from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
Type 7: Vertical Translation Upwards
\( \displaystyle y = \frac{1}{x}+d, \ d \gt 0 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x}+1} \) is translated by \( 1 \) unit upwards from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
Type 8: Vertical Translation Downwards
\( \displaystyle y = \frac{1}{x}-d, \ d \gt 0 \)
The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x}-1} \) is translated by \( 1 \) unit downwards from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).
Type 9: Squaring the Variable
\( \displaystyle y = \frac{1}{x^2} \)
The range of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x^2}} \) is \( y \gt 0 \) as \( x^2 \gt 0, \ x \ne 0 \).
Note that \( \displaystyle \require{AMSsymbols} \color{red}{\frac{1}{x^2}} \gt \color{blue}{\frac{1}{x}} \) for \( 0 \lt x \lt 1 \), and \( \displaystyle \require{AMSsymbols} \color{red}{\frac{1}{x^2}} \lt \color{blue}{\frac{1}{x}} \) for \( x \gt 1 \)
Also \( \displaystyle \require{AMSsymbols} \frac{1}{x^2} = \frac{1}{(-x)^2} \), so the graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x^2}} \) is \(y \)-axis symmetric.
Type 10: Denominators in Standard Form
\( \displaystyle y = \frac{1}{ax+b} \)
For instance, \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{2x+6} = \frac{1}{2(x+3)}} \) is shifted by \( 3 \) units to the left from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{2x}} \) .
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this is soooo helpful! thanks so much!
Thank-you for your comment 🙂
Thanks for posting, this is great! The graphs are really nice too! What would the transformation be if the variable was squared?
Hi Mira.
Thanks for reaching out. Good question, Mira. your question was answered at the bottom of the post.
let us know if you have any further questions 🙂
what are the transformations for when the denominator is in standard form? i mean, how does the ‘b’ value affect the graph?
Hi Lilyana.
Thanks for reaching out.
While we are not 100% sure what “b” is, but assuming the denominator is in a standard form ax + b, some additional explanation has been added at the bottom of the post, so please take a look at it. Feel free to let us know if you have further questions on this.
Regards. iitutor team
I think you did wrong with the horizontal stretch/compression and the vertical stretch/compression, you write the vertical stretch as horizontal stretch, you upside down from type 1 to type 4. I will lose 1 mark from it:(
Hi Cyro.
Thanks for reaching out.
yes, the previous workings may mislead somehow, so we rewrote the topic to provide a better clarification for the translation. So please take a look at the amended theory and let us know what your thoughts are. Sorry for the confusion of the previous writing and hope this clears the most of the translation of the rational functions.