Must-Know 10 Basic Translations of Rational Functions Explained

\( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{0.5}{x}} \) is compressed vertically from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{0.5}{x}} \lt \color{blue}{y = \frac{1}{x}} \)

\( \displaystyle y = \frac{a}{x}, \ a \gt 1 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{2}{x}} \) is stretched vertically from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{2}{x}} \gt \color{blue}{y = \frac{1}{x}} \)

\( \displaystyle y = \frac{1}{bx}, \ 0 \lt b \lt 1 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{0.5x}} \) is stretched horizontally from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{0.5x} = \frac{2}{x}} \gt \color{blue}{y = \frac{1}{x}} \)

\( \displaystyle y = \frac{1}{bx}, \ b \gt 1 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{2x}} \) is compressed horizontally from the graph of \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{2x}} \lt \color{blue}{y = \frac{1}{x}} \)

\( \displaystyle y = \frac{1}{x-c}, \ c \gt 0 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x-1}} \) is translated by \( 1 \) unit to the right from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle y = \frac{1}{x+c}, \ c \gt 0 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x+1}} \) is translated by \( 1 \) unit to the left from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle y = \frac{1}{x}+d, \ d \gt 0 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x}+1} \) is translated by \( 1 \) unit upwards from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle y = \frac{1}{x}-d, \ d \gt 0 \)

The graph of \( \displaystyle \require{AMSsymbols} \color{red}{y = \frac{1}{x}-1} \) is translated by \( 1 \) unit downwards from \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{x}} \).

\( \displaystyle y = \frac{1}{x^2} \)

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.

\( \displaystyle y = \frac{1}{ax+b} \)

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 \( \displaystyle \require{AMSsymbols} \color{blue}{y = \frac{1}{2x}} \) .