A rational function is a function of the form:
$$f(x) = \frac{p(x)}{q(x)}$$
where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.
The graph of a rational function can be sketched by following these steps:
- Find the intercepts.
- Determine the asymptotes.
- Plot additional points.
- Sketch the graph.
The asymptotesof a rational function are linesthat the graph approachesbut nevertouches.
Intercepts
\$x\$-intercepts
The \$x\$-intercepts of a rational function are the solutions to the equation $p(x) = 0$.
\$y\$-intercept
The \$y\$-intercept of a rational function is the value of the function when $x = 0$.
Asymptotes
Vertical Asymptotes
The vertical asymptotes of a rational function are the values of $x$ for which $q(x) = 0$.
Horizontal Asymptotes
The horizontal asymptotes of a rational function are determined by the degrees of the polynomials $p(x)$ and $q(x)$.
- If the degree of $p(x)$ is less than the degree of $q(x)$, the horizontal asymptote is $y = 0$.
- If the degree of $p(x)$ is equal to the degree of $q(x)$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of $p(x)$ and $q(x)$, respectively.
- If the degree of $p(x)$ is greater than the degree of $q(x)$, there is no horizontal asymptote.
If the degreeof $p(x)$ is exactly onegreaterthan the degreeof $q(x)$, the function has a slant asymptote.
Plotting Additional Points
To get a better idea of the shape of the graph, plot additional points by choosing values of $x$ and calculating the corresponding values of $f(x)$.
Sketching the Graph
Connect the points with a smooth curve, making sure to approach the asymptotes without crossing them.
1. Sketch the graph of $f(x) = \frac{x^2 - 1}{x + 1}$. 2. Sketch the graph of $f(x) = \frac{x^2 + 2x + 1}{x^2 - 1}$.
