Writing a Rational Function from Intercepts and Asymptotes

The graph appears to have x-intercepts at $x=\mathrm{–2}$ and $x=3.$ At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asymptotes. The one at $x=\mathrm{–1}$ 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.

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

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

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