Lesson 2: Graphing Rational Functions

Property

The parent rational function is $f(x) = \frac{1}{x}$. An asymptote is a line that the graph of a function approaches but never touches. For this function, the y-axis ($x=0$) is a vertical asymptote, and the x-axis ($y=0$) is a horizontal asymptote.

Examples

Property

For the transformation $g(x) = \frac{a}{x}$ where $a \neq 0$:

• Same asymptotes as parent function: $x = 0$ and $y = 0$
• When $|a| > 1$: vertical stretch by factor of $|a|$
• When $0 < |a| < 1$: vertical compression by factor of $|a|$
• When $a < 0$: reflection across the x-axis

Examples

Property

For rational functions of the form $y = \frac{a}{x-h} + k$:

• Vertical asymptote: $x = h$
• Horizontal asymptote: $y = k$
• The graph is the parent function $f(x) = \frac{1}{x}$ translated $h$ units horizontally and $k$ units vertically

Examples