MA001: College Algebra

Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an $x$-intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity). As a result, we can form a numerator of a function whose graph will pass through a set of $x$-intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.

WRITING RATIONAL FUNCTIONS FROM INTERCEPTS AND ASYMPTOTES

If a rational function has $x$-intercepts at $x=x_{1}, x_{2}, \ldots, x_{n}$, vertical asymptotes at $x=v_{1}, v_{2}, \ldots, v_{m}$ and no $x_{i}=\operatorname{any} v_{j}$, then the function can be written in the form:

$f(x)=a \frac{\left(x-x_{1}\right)^{p_{1}}\left(x-x_{2}\right)^{p_{2}} \cdots\left(x-x_{n}\right)^{p_{n}}}{\left(x-v_{1}\right)^{q_{1}}\left(x-v_{2}\right)^{q_{2}} \cdots\left(x-v_{m}\right)^{q_{n}}}$

where the powers $p_{i}$ or $q_{i}$ on each factor can be determined by the behavior of the graph at the corresponding intercept or asymptote, and the stretch factor a can be determined given a value of the function other than the $x$-intercept or by the horizontal asymptote if it is nonzero.

HOW TO

Given a graph of a rational function, write the function.

EXAMPLE 12

Writing a Rational Function from Intercepts and Asymptotes

Write an equation for the rational function shown in Figure 22.

Figure 22

Solution

The graph appears to have $x$-intercepts at $x=–2$ and $x=3$. At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asymptotes. The one at $x=–1$ seems to exhibit the basic behavior similar to $\frac{1}{x}$, with the graph heading toward positive infinity on one side and heading toward negative infinity on the other. The asymptote at $x=2$ is exhibiting a behavior similar to $\frac{1}{x^{2}}$, with the graph heading toward negative infinity on both sides of the asymptote. See Figure 23.

Figure 23

We can use this information to write a function of the form

$f(x)=a \frac{(x+2)(x-3)}{(x+1)(x-2)^{2}}$

To find the stretch factor, we can use another clear point on the graph, such as the $y$-intercept $(0,–2)$.

This gives us a final function of $f(x)=\frac{4(x+2)(x-3)}{3(x+1)(x-2)^{2}}$.