Section 8.4 Short-Run Behavior of Rational Functions

Supplemental Videos.

The main topics of this section are also presented in the following videos:

Short-Run Behavior of Rational Functions

Subsection 8.4.1 Vertical Asymptotes and Holes

We have previously seen that a polynomial function is defined for all values of \(x\text{,}\) and its graph is a smooth curve without any breaks or holes. The graph of a rational function, on the other hand, will have breaks or holes at those \(x\)-values where it is undefined.

Example 8.26.

Investigate the graph of \(f(x) = \displaystyle{\frac{2}{(x - 3)^2}}\) near \(x = 3\text{.}\)

Solution.

This function is undefined for \(x = 3\text{,}\) so there is no point on the graph with \(x\)-coordinate \(3\text{.}\) However, we can make a table of values for other values of \(x\text{.}\) Plotting the ordered pairs in the table results in the points shown in Figure 8.27.

\(x\)	\(y\)
\(0\)	\(\dfrac{2}{9}\)
\(1\)	\(\dfrac{1}{2}\)
\(2\)	\(2\)
\(3\)	undefined
\(4\)	\(2\)
\(4\)	\(\dfrac{1}{2}\)
\(6\)	\(\dfrac{2}{9}\)

described in detail following the image — Figure 8.27.

Next, we make a table showing \(x\)-values close to \(3\text{,}\) as in Figure 8.28a. As we choose \(x\)-values closer and closer to \(3\text{,}\) \((x - 3)^2\) gets closer to \(0\text{,}\) so the fraction \(\dfrac{2}{(x-3)^2}\) gets very large. This means that the graph approaches, but never touches, the vertical line \(x=3\text{.}\) In other words, the graph has a vertical asymptote at \(x=3\text{.}\) We indicate the vertical asymptote by a dashed line, as shown in Figure 8.28b.

The vertical line \(x=3\) in the example above is an example of a vertical asymptote. In general, we have the following result.

Vertical Asymptotes.

If \(Q(a) = 0\) but \(P(a) \ne 0\text{,}\) then the graph of the rational function \(f(x) = \displaystyle{\frac{P(x)}{Q(x)}}\) has a vertical asymptote at \(x=a\text{.}\)

However, if both the numerator and denominator of the rational function share a root, we no longer call this a vertical asymptote.

If \(P(a)\) and \(Q(a)\) are both zero in the rational function \(f(x)=\frac{P(x)}{Q(x)}\text{,}\) but the reduced expression for \(f(x)\) is defined at \(x=a\text{,}\) then the graph of \(f(x)\) has a hole at \(x=a\) rather than an asymptote.

Example 8.29.

Find the vertical asymptotes of \(G(x) = \displaystyle{\frac{4x^2}{x^2 - 4}}\text{.}\)

Solution.

To find the vertical asymptotes, we are looking for values of \(a\) such that \(4a^2 \neq 0\) and \(a^2-4=0\text{.}\) The equation \(a^2-4=0\) has solutions \(a=2,-2\text{,}\) and both \(4(\alert{2})^2\neq0\) and \(4(\alert{-2})^2\neq0\text{.}\) This means that there are two vertical asymptotes, namely \(x=2\) and \(x=-2\text{.}\)

Near a vertical asymptote, the graph of a rational function has one of the four characteristic shapes, illustrated in Figure 8.30. Locating the vertical asymptotes can help us make a quick sketch of a rational function.

Example 8.31.

Locate the vertical asymptotes and sketch the graph of \(g(x) = \displaystyle{\frac{x}{x + 1}}\text{.}\)

Solution.

The denominator, \(x+1\text{,}\) equals zero when \(x = -1\text{.}\) Because the numerator does not equal zero when \(x = -1\text{,}\) there is a vertical asymptote at \(x = -1\text{.}\) The asymptote separates the graph into two pieces.

We can use the Table feature of a calculator to evaluate \(g(x)\) for several values of \(x\) on either side of the asymptote, as shown in Figure 8.32a. We plot the points found in this way; then connect the points on either side of the asymptote to obtain the graph shown in Figure 8.32b.

Example 8.33.

Find the vertical asymptotes of \(f(x) = \displaystyle{\frac{1}{x^2 - 4}}\) . Locate any \(x\)-intercepts.
Evaluate the function at \(x = -3\text{,}\) \(-1\text{,}\) \(1\text{,}\) and \(3\text{.}\) Sketch a graph of the function.

Solution.

The denominator, \(x^2-4\text{,}\) equals zero when \(x=2\) and \(x=-2\text{.}\) Because the numerator does not equal zero for either of these, there are two vertical asymptotes, at \(x=2\) and \(x=-2\text{.}\)

To find the \(x\)-intercepts, we are looking for values of \(x\) such that the function is equal to zero:

\begin{equation*} \dfrac{1}{x^2-4} = 0 \end{equation*}

Since \(1\) divided by any real number is nonzero, there is no such \(x\text{.}\) This means that there are no \(x\)-intercepts.
\(x\) \(-3\) \(-1\) \(1\) \(3\)

\(y\) \(\frac{1}{5} \) \(\frac{-1}{3} \) \(\frac{-1}{3} \) \(\frac{1}{5} \)

rational function

Exercises 8.4.2 Exercises

1. Long- and Short-Run Behavior.

Describe the long-run behavior of the following functions, and identify horizontal asymptotes.

a) \(g(x)=\dfrac{3 x^{2}}{x^{2}-1}\)

As \(x\) goes to \(+\infty\text{,}\) \(g(x)\) goes to .

As \(x\) goes to \(-\infty\text{,}\) \(g(x)\) goes to .

Horizontal asymptote: \(y =\)

b) \(f(x)=\dfrac{5}{(x- 3 )^{4}}\)

As \(x\) goes to \(+\infty\text{,}\) \(f(x)\) goes to .

As \(x\) goes to \(-\infty\text{,}\) \(f(x)\) goes to .

Horizontal asymptote: \(y =\)

Hint.

Recall a function has a horizontal asymptote \(y=k\text{,}\) for some constant \(k\) if the long-run behavior approaches the value \(k\text{.}\) In other words, if as \(x\to \infty\) or \(x\to-\infty\text{,}\) \(f(x)\) approaches \(k\text{,}\) then the line \(y=k\) is a horizontal asymptote.

The long-run behavior of a rational function can be found by dividing the leading terms of the numerator and denominator and finding the long-run behavior of the resulting power function.

2. Short-Run Behavior.

In this problem, we will examine the short-run behavior of the following functions. As you complete the chart, look carefully for patterns.

a) \(f(x)= \dfrac{x-5}{x + 7}\)

b) \(g(x)= \dfrac{\left(x+3\right)\left(x-6\right)}{x^2-1}\)

c) \(h(x)= \dfrac{2\left(x + 1\right)\left(x-7\right)}{x + 1}\)

d) \(\ell(x)= \dfrac{\left(x + 2.8\right)\left(x + 9\right)}{\left(x+1\right)\left(x + 9 \right)}\)

e) \(m(x)= \dfrac{x^2-6x+8}{x-2}\)

Complete the following table using the formulas for each of the above functions and any additional information you can obtain from the graph of each function. For this question, consider holes and vertical asymptotes to be undefined points.

function	number of (distinct) zeros	number of undefined points
\(f(x)\)
\(g(x)\)
\(h(x)\)
\(\ell(x)\)
\(m(x)\)

Function features

b) List the roots (in form \(x = r\)) in order from smallest to largest. Also, list the undefined points of the function from smallest to largest. Finally, list the vertical asymptote(s). If there are more blanks provided than needed for a given feature, enter NONE in all remaining, unneeded blanks. The information for function \(f(x)\) has been completed as an example.

\(f(x)\text{:}\)

Zeros: \(x = 4\text{,}\) \(x =\)NONE

Holes: \(x =\) NONE, \(x=\) NONE

Vertical Asymptotes: \(x =-6\text{,}\) \(x=\) NONE

\(g(x)\text{:}\)

Zeros: \(x =\) , \(x =\)

Holes: \(x =\) , \(x=\)

Vertical Asymptotes: \(x =\) , \(x=\)

\(h(x)\text{:}\)

Zeros: \(x =\) , \(x =\)

Holes: \(x =\) , \(x=\)

Vertical Asymptotes: \(x =\) , \(x=\)

\(\ell(x)\text{:}\)

Zeros: \(x =\) , \(x =\)

Holes: \(x =\) , \(x=\)

Vertical Asymptotes: \(x =\) , \(x=\)

\(m(x)\text{:}\)

Zeros: \(x =\) , \(x =\)

Holes: \(x =\) , \(x=\)

Vertical Asymptotes: \(x =\) , \(x=\)

Hint.

A rational function \(f(x)=\dfrac{P(x)}{Q(x)}\) has a zero at \(x=a\) if \(P(a)=0\) but \(Q(a)\neq 0\text{.}\)

Undefined points occur when the denominator is undefined. In other words, for all \(x\) values such that \(Q(x)=0\text{.}\)

Recall the following definitions to help you answer the remaining questions:

If \(Q(a) = 0\) but \(P(a) \neq 0\text{,}\) then the graph of the rational function \(f(x) = \dfrac{P(x)}{Q(x)}\) has a vertical asymptote at \(x = a\text{.}\)

If \(Q(a) = 0\) and \(P(a) = 0\) in the expression \(f(x) = \dfrac{P(x)}{Q(x)}\) but the reduced expression of \(f(x)\) is defined at \(x = a\text{,}\) then the graph of \(f(x)\) has a hole at \(x = a\text{.}\)

3. All Behavior of a Rational Function.

Let \(r(x) = \dfrac{x^2-x-2}{x^2+4x+3}\text{.}\)

a) Describe the long-run behavior or \(r(x)\text{,}\) and give any horizontal asymptotes of \(r(x)\text{.}\) Enter NONE in the blanks provided if there are no horizontal asymptotes.

As \(x\) goes to both positive and negative infinity, \(r(x)\) goes to 1
As \(x\) goes to both positive and negative infinity, \(r(x)\) goes to positive infinity
As \(x\) goes to both positive and negative infinity, \(r(x)\) goes to 0
As \(x\) goes to both positive and negative infinity, \(r(x)\) goes to negative infinity
As \(x\) goes to positive infinity, \(r(x)\) goes to positive infinity, and as \(x\) goes to negative infinity, \(r(x)\) goes to negative infinity

\(y=\)

b) What are the zeros of \(r(x)\text{?}\) Enter NONE in any remaining blanks.

\(x =\)

c) Determine whether \(r(x)\) has vertical asymptotes. List them. Enter NONE in any remaining blanks.

Yes, the given rational function has a vertical asymptote
No, the given rational function does not have a vertical asymptote

\(x =\)

d) Does \(r(x)\) have a hole in its graph? If so, find the coordinates of the hole. Otherwise, enter NONE in the blanks provided.

Yes, the given rational function has a hole
No, the given rational function does not have a hole

\((\) , \()\)

Hint.

The long-run behavior of a rational function can be found by dividing the leading terms of the numerator and denominator and finding the long-run behavior of the resulting power function.

A rational function \(f(x)=\dfrac{P(x)}{Q(x)}\) has a zero at \(x=a\) if \(P(a)=0\) but \(Q(a)\neq 0\text{.}\)

Undefined points occur when the denominator is undefined. In other words, for all \(x\) values such that \(Q(x)=0\text{.}\)

Recall the following definitions to help you answer the remaining questions:

If \(Q(a) = 0\) but \(P(a) \neq 0\text{,}\) then the graph of the rational function \(f(x) = \dfrac{P(x)}{Q(x)}\) has a vertical asymptote at \(x = a\text{.}\)

4. Identifying Holes.

Consider the function \(f(x) = \dfrac{x^3-x}{x-1}\text{,}\) whose graph is a parabola with a hole in it. Give the equation of this parabola, \(f_2 (x)\text{,}\) as well as the coordinates of the hole.

\(f_2 (x) =\)

Coordinates of hole: \((\) , \()\)

Hint.

Try factoring the numerator to simplify the equation. Recall there’s a hole in the graph of the function where the numerator and denominator share a zero. Find the \(x\) value where the hole will occur.

After simplifying the function, evaluate at the \(x\)-value where the hole occurs to find the point. Note that the simplified function should be a quadratic function.

5. Finding Formula from Graph.

Determine a possible formula of least degree for the rational function \(g(x)\) whose graph is shown below.

\(g(x) =\)

Hint.

The long-run behavior of a rational function can be found by dividing the leading terms of the numerator and denominator and finding the long-run behavior of the resulting power function.

A rational function \(f(x)=\dfrac{P(x)}{Q(x)}\) has a zero at \(x=a\) if \(P(a)=0\) but \(Q(a)\neq 0\text{.}\)

Undefined points occur when the denominator is undefined. In other words, for all \(x\) values such that \(Q(x)=0\text{.}\)

Recall the following definitions to help you answer the remaining questions:

If \(Q(a) = 0\) but \(P(a) \neq 0\text{,}\) then the graph of the rational function \(f(x) = \dfrac{P(x)}{Q(x)}\) has a vertical asymptote at \(x = a\text{.}\)

Determine from the graph all of these features: horizontal and vertical asymptotes and \(x\)-intercepts. Use the information above to figure out where each of these pieces should go in the formula. For example, if \(x=a\) is a zero of the function, we would include the factor \((x-a)\) in the numerator of the function, but not in the denominator.

6. All Behavior of a Rational Function.

Determine the long-run behavior of the following rational functions. Then, determine their \(x\)-intercepts, \(y\)-intercepts, and horizontal/vertical asymptotes, if there are any. Write NONE in any unused blanks. If there is more than one root, list the \(x\)-intercepts in order from least \(x\) coordinate to greatest. Similarly, if there is more than one vertical asymptote, list the asymptotes in order from least to greatest.

a) \(y = \dfrac{x + 1}{x + 5}\)

Long-run behavior:

As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 1
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 0
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to \(+\infty\)
As \(x\) goes to \(+ \infty\text{,}\) \(y\) goes to \(+ \infty\text{,}\) and as \(x\) goes to \(-\infty\text{,}\) \(y\) goes to \(- \infty\)

\(x\)-intercepts:

\((\) , \()\)

\(y\)-intercept:

\((\) , \()\)

Horizontal asymptote: \(y=\)

Vertical asymptote(s):

\(x =\)

b) \(y = \dfrac{x + 3}{(x-1)^2}\)

Long-run behavior:

As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 1
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 0
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to \(+\infty\)
As \(x\) goes to \(+ \infty\text{,}\) \(y\) goes to \(+ \infty\text{,}\) and as \(x\) goes to \(-\infty\text{,}\) \(y\) goes to \(- \infty\)

\(x\)-intercepts:

\((\) , \()\)

\(y\)-intercept:

\((\) , \()\)

Horizontal asymptote: \(y=\)

Vertical asymptote(s):

\(x =\)

c) \(h(x) = \dfrac{x^2-4}{x^3+4x^2}\)

Long-run behavior:

As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 1
As \(x\) goes to \(+ \infty\text{,}\) \(y\) goes to \(+ \infty\text{,}\) and as \(x\) goes to \(-\infty\text{,}\) \(y\) goes to \(- \infty\)
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 0
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to \(+\infty\)

\(x\)-intercepts:

\((\) , \()\)

\(y\)-intercept:

\((\) , \()\)

Horizontal asymptote: \(y=\)

Vertical asymptote(s):

\(x =\)

d) \(k(x) = \dfrac{x^2-9}{x^2-3x+2}\)

Long-run behavior:

As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 1
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to \(+\infty\)
As \(x\) goes to \(+ \infty\text{,}\) so does \(y\text{,}\) and as \(x\) goes to \(-\infty\text{,}\) so does \(y\)
As \(x\) goes to both \(+ \infty\) and \(-\infty\text{,}\) \(y\) goes to 0

\(x\)-intercepts:

\((\) , \()\)

\(y\)-intercept:

\((\) , \()\)

Horizontal asymptote: \(y=\)

Vertical asymptote(s):

\(x =\)

Hint.

The long-run behavior of a rational function can be found by dividing the leading terms of the numerator and denominator and finding the long-run behavior of the resulting power function.

A rational function \(f(x)=\dfrac{P(x)}{Q(x)}\) has a zero at \(x=a\) if \(P(a)=0\) but \(Q(a)\neq 0\text{.}\)

Undefined points occur when the denominator is undefined. In other words, for all \(x\) values such that \(Q(x)=0\text{.}\)

Recall the following definitions to help you answer the remaining questions:

If \(Q(a) = 0\) but \(P(a) \neq 0\text{,}\) then the graph of the rational function \(f(x) = \dfrac{P(x)}{Q(x)}\) has a vertical asymptote at \(x = a\text{.}\)

The \(y\)-intercept occurs when \(x=0\text{,}\) so you can evaluate the function at \(x=0\text{.}\)

7. Finding Formula from Graph.

Determine which of the following functions matches the given graph.

\(\displaystyle y= \dfrac{x-3}{(x+3)(x + 5)}\)
\(\displaystyle y= \dfrac{3-x}{(x-3)(x + 5)}\)
\(\displaystyle y= \dfrac{x-3}{x + 5}\)
\(\displaystyle y= \dfrac{x-3}{(x-3)(x + 5)}\)

Hint.

The long-run behavior of a rational function can be found by dividing the leading terms of the numerator and denominator and finding the long-run behavior of the resulting power function.

A rational function \(f(x)=\dfrac{P(x)}{Q(x)}\) has a zero at \(x=a\) if \(P(a)=0\) but \(Q(a)\neq 0\text{.}\)

Undefined points occur when the denominator is undefined. In other words, for all \(x\) values such that \(Q(x)=0\text{.}\)

Recall the following definitions to help you answer the remaining questions:

If \(Q(a) = 0\) but \(P(a) \neq 0\text{,}\) then the graph of the rational function \(f(x) = \dfrac{P(x)}{Q(x)}\) has a vertical asymptote at \(x = a\text{.}\)

You can either try to write a formula for the function in the graph or find the graph features in each formula option given. Either method will help you find the correct answer.

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\(x\)	\(-3\)	\(-1\)	\(1\)	\(3\)
\(y\)	\(\frac{1}{5} \)	\(\frac{-1}{3} \)	\(\frac{-1}{3} \)	\(\frac{1}{5} \)