8.4: Short-Run Behavior of Rational Functions

Prior Lesson

objectives

• Find zeros and vertical asymptote of rational functions
• Find holes of and graph rational functions
• Find a possible formula for a rational function, given its graph.

Lesson Guide

Warm-Up

Have students do Problem 1.

Find zeros and vertical asymptote of rational functions

For a rational function $r(x)=\frac{p(x)}{q(x)}$ in simplified form 
* the zeros of $r(x)$ are the same as the zeros of the numerator $p(x)$ (provided they are in the domain of $r(x)$)
* the graph of $r(x)$ has a vertical asymptote at each of the zeros of the denominator $q(x)$.

Give an example of a rational function and find its zeros and vertical asymptote.

• Example:

Have students do Problem 2. Assign each group one or two functions to work on. Either project or redraw the table on the whiteboard and have each group fill in the row they worked on.

Once the table is filled in, ask students to see if they can find relationships between the zeros, the undefined points, and the vertical asymptotes.

Have students do Problem 3.

Find holes of and graph rational functions

When a the numerator and denominator of a rational function $r(x)=\frac{p(x)}{q(x)}$ have common zeros,
these zeros can give holes in the graph of the rational function $r(x)$.

Do a simple example, like $h(x)=\frac{x^2+x-6}{x+3}$, to demonstrate how to graph a rational function and find any holes. Be sure to first factor the numerator in order to find common zeros. To graph $h(x)$, first simplify, then graph the simplified function but indicate that there is a hole at $x=-3$ with an open circle.

• Example:

Have students do Problems 4 and 5.

Find a possible formula for a rational function, given its graph.

Do an example of how to determine a possible formula for a rational function, given its graph.

• Example:

Have students do Problems 6 and 7.

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