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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 4.7 Properties and Graphs of Rational Functions (PR7)

Subsection 4.7.1 Activities

Definition 4.7.1.

A function \(r\) is rational provided that it is possible to write \(r\) as the ratio of two polynomials, \(p\) and \(q\text{.}\) That is, \(r\) is rational provided that for some polynomial functions \(p\) and \(q\text{,}\) we have
\begin{equation*} r(x) = \dfrac{p(x)}{q(x)}\text{.} \end{equation*}
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Observation 4.7.2.

Rational functions occur in many applications, so our goal in this lesson is to learn about their properties and be able to graph them. In particular we want to investigate the domain, end behavior, and zeros of rational functions.
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Activity 4.7.3.

Consider the rational function
\begin{equation*} r(x) = \dfrac{x^2-3x+2}{x^2-4x+3}\text{.} \end{equation*}
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(a)
Find \(r(1) \text{,}\) \(r(2) \text{,}\) \(r(3)\text{,}\) and \(r(4)\text{.}\)
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Answer.
\(r(1)\) and \(r(3)\) do not exist. \(r(2)=0\) \(r(4)=2\)
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(b)
Label each of these four values as giving us information about the DOMAIN of \(r(x)\text{,}\) information about the ZEROES of \(r(x)\text{,}\) or NEITHER.
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Answer.
\(r(1)\) and \(r(3)\) give information about the domain. \(r(2)\) gives information about the zeroes. \(r(4)\) does not give information about either.
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Observation 4.7.4.

Let \(p\) and \(q\) be polynomial functions so that \(r(x) = \dfrac{p(x)}{q(x)}\) is a rational function. The domain of \(r\) is the set of all real numbers except those for which \(q(x) = 0\text{.}\)
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Activity 4.7.5.

Let’s investigate the domain of \(r(x)\) more closely. We will be using the same function from the previous activity:
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\begin{equation*} r(x) = \dfrac{x^2-3x+2}{x^2-4x+3}\text{.} \end{equation*}
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(a)
Rewrite \(r(x)\) by factoring the numerator and denominator, but do not try to simplify any further. What do you notice about the relationship between the values that are not in the domain and how the function is now written?
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Answer.
\(r(x)=\dfrac{(x-2)(x-1)}{(x-3)(x-1)}\) The values not in the domain are zeros of the denominator.
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(b)
The function was not defined for \(x=3\text{.}\) Make a table for values of \(r(x)\) near \(x=3\text{.}\)
Table 4.7.6.
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\(x\) \(r(x)\)
\(2\)
\(2.9\)
\(2.99\)
\(2.999\)
\(3\) undefined
\(3.001\)
\(3.01\)
\(3.1\)
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Answer.
\(x\) \(r(x)\)
\(2\) \(0\)
\(2.9\) \(-9\)
\(2.99\) \(-99\)
\(2.999\) \(-999\)
\(3\) undefined
\(3.001\) \(1001\)
\(3.01\) \(101\)
\(3.1\) \(11\)
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(c)
Which of the following describe the behavior of the graph near \(x=3\text{?}\)
  1. As \(x\to 3\text{,}\) \(r(x)\) approaches a finite number
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  2. As \(x\to 3\) from the left, \(r(x)\to\infty\)
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  3. As \(x \to 3\) from the left, \(r(x)\to -\infty\)
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  4. As \(x \to 3\) from the right, \(r(x)\to \infty\)
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  5. As \(x \to 3\) from the right, \(r(x)\to -\infty\)
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Answer.
C and D
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(d)
The function was also not defined for \(x=1\text{.}\) Make a table for values of \(r(x)\) near \(x=1\text{.}\)
Table 4.7.7.
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\(x\) \(r(x)\)
\(0\)
\(0.9\)
\(0.99\)
\(0.999\)
\(1\) undefined
\(1.001\)
\(1.01\)
\(1.1\)
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Answer.
\(x\) \(r(x)\)
\(0\) \(0.667\)
\(0.9\) \(0.524\)
\(0.99\) \(0.502\)
\(0.999\) \(0.500\)
\(1\) undefined
\(1.001\) \(0.499\)
\(1.01\) \(0.497\)
\(1.1\) \(0.474\)
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(e)
Which of the following describe the behavior of the graph near \(x=1\text{?}\)
  1. As \(x\to 1\text{,}\) \(r(x)\) approaches a finite number
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  2. As \(x\to 1\) from the left, \(r(x)\to\infty\)
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  3. As \(x \to 1\) from the left, \(r(x)\to -\infty\)
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  4. As \(x \to 1\) from the right, \(r(x)\to \infty\)
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  5. As \(x \to 1\) from the right, \(r(x)\to -\infty\)
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Answer.
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(f)
The function is behaving differently near \(x=1\) than it is near \(x=3\text{.}\) Can you see anything in the factored form of \(r(x)\) that may help you account for the difference?
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Answer.
The \(x-1\) factor occurs in the numerator and denominator.
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Remark 4.7.8. Features of a rational function.

Let \(r(x) = \dfrac{p(x)}{q(x)}\) be a rational function.
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  • If \(p(a) = 0\) and \(q(a) \ne 0\text{,}\) then \(r(a) = 0\text{,}\) so \(r\) has a zero at \(x = a\text{.}\)
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  • If \(q(a) = 0\) and \(p(a) \ne 0\text{,}\) then \(r(a)\) is undefined and \(r\) has a vertical asymptote at \(x = a\text{.}\)
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  • If \(p(a) = 0\) and \(q(a) = 0\) and we can show that there is a finite number \(L\) such that \(r(x) \to L \text{,}\) then \(r(a)\) is not defined and \(r\) has a hole at the point \((a,L)\text{.}\)
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Observation 4.7.9.

Another property of rational functions we want to explore is the end behavior. This means we want to explore what happens to a given rational function \(r(x)\) when \(x\) goes toward positive infinity or negative infinity.
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Activity 4.7.10.

Consider the rational function \(\displaystyle r(x)=\dfrac{1}{x^3}\text{.}\)
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(a)
Plug in some very large positive numbers for \(x\) to see what \(r(x)\) is tending toward. Which of the following best describes the behavior of the graph as \(x\) approaches positive infinity?
  1. As \(x\to \infty\text{,}\) \(r(x)\to \infty\text{.}\)
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  2. As \(x\to \infty\text{,}\) \(r(x)\to -\infty\text{.}\)
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  3. As \(x\to \infty\text{,}\) \(r(x)\to 0\text{.}\)
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  4. As \(x\to \infty\text{,}\) \(r(x)\to 1\text{.}\)
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Answer.
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(b)
Now let’s look at \(r(x)\) as \(x\) tends toward negative infinity. Plug in some very large negative numbers for \(x\) to see what \(r(x)\) is tending toward. Which of the following best describes the behavior of the graph as \(x\) approaches negative infinity?
  1. As \(x\to -\infty\text{,}\) \(r(x)\to \infty\text{.}\)
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  2. As \(x\to -\infty\text{,}\) \(r(x)\to -\infty\text{.}\)
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  3. As \(x\to -\infty\text{,}\) \(r(x)\to 0\text{.}\)
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  4. As \(x\to -\infty\text{,}\) \(r(x)\to 1\text{.}\)
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Answer.
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Observation 4.7.11.

We can generalize what we have just found to any function of the form \(\dfrac{1}{x^n}\text{,}\) where \(n>0\text{.}\) Since \(x^n\) increases without bound as \(x \to \infty\text{,}\) we find that \(\dfrac{1}{x^n}\) will tend to 0. In fact, the numerator can be any constant and the function will still tend to 0!
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Similarly, as \(x \to -\infty\text{,}\) we find that \(\dfrac{1}{x^n}\) will tend to 0 too.
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Activity 4.7.12.

Consider the rational function \(\displaystyle r(x) = \dfrac{3x^2 - 5x + 1}{7x^2 + 2x - 11}\text{.}\)
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Observe that the largest power of \(x\) that’s present in \(r(x)\) is \(x^2\text{.}\) In addition, because of the dominant terms of \(3x^2\) in the numerator and \(7x^2\) in the denominator, both the numerator and denominator of \(r\) increase without bound as \(x\) increases without bound.
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(a)
In order to understand the end behavior of \(r\text{,}\) we will start by writing the function in a different algebraic form.
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Multiply the numerator and denominator of \(r\) by \(\dfrac{1}{x^2}\text{.}\) Then distribute and simplify as much as possible in both the numerator and denominator to write \(r\) in a different algebraic form. Which of the following is that new form?
  1. \(\displaystyle \dfrac{3x^4 - 5x^3 +x^2}{7x^4+2x^3-11x^2}\)
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  2. \(\displaystyle \dfrac{3 - \dfrac{5}{x} + \dfrac{1}{x^2}}{7 + \dfrac{2}{x} - \dfrac{11}{x^2}}\)
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  3. \(\displaystyle \dfrac{\dfrac{3x^2}{x^2} - \dfrac{5x}{x^2} + \dfrac{1}{x^2}}{\dfrac{7x^2}{x^2} + \dfrac{2x}{x^2} - \dfrac{11}{x^2}}\)
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  4. \(\displaystyle \dfrac{3x^2 - 5x +1}{7x^4+2x^3-11x^2}\)
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Answer.
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(b)
Now determine the end behavior of each piece of the numerator and each piece of the denominator.
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Hint.
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Answer.
Numerator goes to \(3\) and denominator goes to \(7\text{.}\)
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(c)
Simplify your work from the previous step. Which of the following best describes the end behavior of \(r(x)\text{?}\)
  1. As \(x \to \pm \infty\text{,}\) \(r(x)\) goes to \(0\text{.}\)
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  2. As \(x \to \pm \infty\text{,}\) \(r(x)\) goes to \(\dfrac{3}{7}\text{.}\)
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  3. As \(x \to \pm \infty\text{,}\) \(r(x)\) goes to \(\infty\text{.}\)
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  4. As \(x \to \pm \infty\text{,}\) \(r(x)\) goes to \(-\infty\text{.}\)
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Answer.
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Observation 4.7.13.

If the end behavior of a function tends toward a specific value \(a\text{,}\) then we say that the function has a horizontal asymptote at \(y=a\text{.}\)
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Activity 4.7.14.

Find the horizontal asymptote (if one exists) of the following rational functions. Follow the same method we used in Activity 4.7.12.
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(a)
\(f(x)=\dfrac{4x^3-3x^2+6}{9x^3+7x-5}\)
  1. \(\displaystyle y=0\)
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  2. \(\displaystyle y=\dfrac{4}{9}\)
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  3. \(\displaystyle y=-\dfrac{3}{7}\)
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  4. \(\displaystyle y=-\dfrac{6}{5}\)
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  5. There is no horizontal asymptote.
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Answer.
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(b)
\(g(x)=\dfrac{4x^3-3x^2+6}{9x^5+7x-5}\)
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  1. \(\displaystyle y=0\)
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  2. \(\displaystyle y=\dfrac{4}{9}\)
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  3. \(\displaystyle y=-\dfrac{3}{7}\)
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  4. \(\displaystyle y=-\dfrac{6}{5}\)
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  5. There is no horizontal asymptote.
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Answer.
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(c)
\(h(x)=\dfrac{4x^5-3x^2+6}{9x^3+7x-5}\)
  1. \(\displaystyle y=0\)
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  2. \(\displaystyle y=\dfrac{4}{9}\)
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  3. \(\displaystyle y=-\dfrac{3}{7}\)
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  4. \(\displaystyle y=-\dfrac{6}{5}\)
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  5. There is no horizontal asymptote.
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Answer.
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Activity 4.7.15.

Some patterns have emerged from the previous problem. Fill in the rest of the sentences below to describe how to find horizontal asymptotes of rational functions.
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(a)
If the degree of the numerator is the same as the degree of the denominator, then...
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Answer.
the horizontal asymptote is a non-zero number.
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(b)
If the degree of the numerator is less than the degree of the denominator, then...
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Answer.
there is a horizontal asymptote at \(0\text{.}\)
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(c)
If the degree of the numerator is greater than the degree of the denominator, then...
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Answer.
there is no horizontal asymptote.
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Activity 4.7.16.

Consider the following six graphs of rational functions:
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(a)
Which of the graphs above represents the function \(\displaystyle f(x)=\dfrac{2x}{x^2-2x-8}\text{?}\)
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Answer.
Graph A
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(b)
Which of the graphs above represents the function \(\displaystyle g(x)=\dfrac{x^2+3}{2x^2-8}\text{?}\)
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Answer.
Graph E
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Activity 4.7.17.

Consider the following six graphs of rational functions:
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(a)
Which of the graphs above represents the function \(\displaystyle f(x)=\dfrac{x^2+5x+6}{(x^2-3x-10)(x+4)}\text{?}\)
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Answer.
Graph A
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(b)
Which of the graphs above represents the function \(\displaystyle g(x)=\dfrac{x^2-3x-10}{(x+2)(x^2-x-20)}\text{?}\)
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Answer.
Graph E
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Activity 4.7.18.

Let \(f(x)=\displaystyle \dfrac{ -{\left(x - 1\right)} {\left(x - 4\right)} }{ 2 \, {\left(x + 3\right)}^{2} {\left(x - 1\right)} }\text{.}\)
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(b)
Find the \(y\)-intercept of the graph of \(f(x)\text{.}\)
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Answer.
\(\left(0,\dfrac{2}{9}\right)\)
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(c)
Find any horizontal asymptotes on the graph of \(f(x)\text{.}\)
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Answer.
\(y=0\)
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(d)
Find any vertical asymptotes on the graph of \(f(x)\text{.}\)
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Answer.
\(x=-3\text{.}\)
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(e)
Find any holes on the graph of \(f(x)\text{.}\)
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Answer.
There is a hole at \(\left(1, \dfrac{3}{32} \right)\text{.}\)
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Activity 4.7.19.

For each of the following rational functions, identify the location of any potential hole in the graph. Then, create a table of function values for input values near where the hole should be located. Use your work to decide whether or not the graph indeed has a hole, with written justification.
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(a)
\(\displaystyle r(x) = \dfrac{x^2-16}{x+4}\)
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Answer.
Hole when \(x=-4\text{.}\)
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(b)
\(\displaystyle s(x) = \dfrac{(x-2)^2(x+3)}{x^2 - 5x + 6}\)
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Answer.
Hole when \(x=2\text{.}\) The graph near \(x=2\) behaves like \(x=2\) is a root with odd multiplicity.
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(c)
\(\displaystyle u(x) = \dfrac{(x-2)^3(x+3)}{x^2 - 5x + 6}\)
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Answer.
Hole when \(x=2\text{.}\) The graph near \(x=2\) behaves like \(x=2\) is a root with even multiplicity.
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(d)
\(\displaystyle w(x) = \dfrac{(x-2)(x+3)}{(x^2 - 5x + 6)^2}\)
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Answer.
Vertical asymptote at \(x=2\text{.}\)
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Activity 4.7.20.

Suppose you are given a function \(r(x) = \dfrac{p(x)}{q(x)}\text{,}\) and you know that \(p(3) = 0\) and \(q(3) = 0\text{.}\) What can you conclude about the function \(r(x)\) at \(x = 3\text{?}\)
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  1. \(r(x)\) has a hole at \(x=3\text{.}\)
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  2. \(r(x)\) has an asymptote at \(x=3\text{.}\)
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  3. \(r(x)\) has either a hole or an asymptote at \(x=3\text{.}\)
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  4. \(r(x)\) has neither a hole nor an asymptote at \(x=3\text{.}\)
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Answer.
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Subsection 4.7.2 Exercises

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