Lesson 4 Are You Rational? Solidify Understanding

Jump Start

1.

Select the numbers that are rational numbers.

A.

$3$

B.

$- 5$

C.

$\frac{2}{3}$

D.

$\frac{20}{3}$

E.

$14$

F.

$2.7$

G.

$\sqrt{5}$

H.

$2^{3}$

I.

$3^{- 3}$

J.

$\log_{2} 9$

K.

$\frac{7}{0}$

2.

Write your definition of a rational number.

3.

Write two examples of rational functions.

4.

Compare rational functions and rational numbers. How are they alike? How are they different?

Learning Focus

Write equivalent rational expressions.

Find the features of rational functions with numerators that are one degree greater than the denominator.

What does the graph of a rational function look like if the function has common factors in the numerator and denominator?

What does the graph of a rational function look like if the degree of the numerator is greater than the degree of the denominator?

Open Up the Math: Launch, Explore, Discuss

The formal definition of a rational function is as follows:

A function $f (x)$ is called a rational function if and only if it can be written in the form $f (x) = \frac{P (x)}{Q (x)}$ where $P$ and $Q$ are polynomials in $x$ and $Q$ is not the zero polynomial.

When working with polynomials previously, it was useful to draw connections between polynomials and integers. In this task, we will use connections between rational numbers and rational expressions and functions to help us to think about operations on rational expressions and functions.

We are going to start by using what we know about rational numbers to perform operations on rational expressions. The first thing we often need to do is to divide common factors from the numerator and denominator. This operation changes the form of the number or expression, but does not change the value. For instance, a common factor of $2$ can be divided from the numerator and denominator of $\frac{2}{4}$ to get $\frac{1}{2}$ , but as the diagram shows, these are just two different ways of expressing the same amount.

1.

Fill in any missing parts of the fractions or empty cells in the table. Start at the top of each problem and work your way down one step at a time.

	A	B	C
Given:	$\frac{24}{30}$	$\frac{x^{2} - x - 6}{x^{2} - 4}$	$\frac{x^{2} + 8 x + 15}{x^{2} + 9 x + 18}$
Look for common factors:	$\frac{2 \cdot 2 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 5}$	$\frac{}{(x + 2) (x - 2)}$
Divide numerator and denominator by the same factor(s):	$\frac{2 \cdot 2 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 5}$
Write an equivalent expression:	$\frac{4}{5}$	$\frac{x - 3}{x - 2}$	$\frac{x + 5}{x + 6}$

2.

Why does dividing the numerator and denominator by the same factor keep the value of the expression the same?

3.

If you were given the expression $\frac{x}{x^{2} - 1}$ , would it be equivalent if rewritten like this: $\frac{x}{x^{2} - 1} = \frac{1}{x - 1}$ ?

Explain your answer.

Previously, we learned to predict vertical and horizontal asymptotes, and to find intercepts for graphing rational functions.

4.

Given $f (x) = \frac{x^{2} - x - 6}{x^{2} - 4}$ , predict the vertical and horizontal asymptotes and find the intercepts.

5.

Use technology to view the graph. Were your predictions correct? What occurs on the graph at $x = - 2$ ?

Sometimes the numerator of a fraction is greater than the denominator. A rational expression is similar, except that instead of comparing the numeric value of the numerator and denominator, the comparison is based on the degree of each polynomial.

As you may remember, fractions where the numerator is greater than the denominator can be rewritten in an equivalent form called a mixed number. If the numerator is greater than the denominator then we divide the numerator by the denominator and write the remainder as a fraction. In math terms we would say:

If $a > b$ , then the fraction $\frac{a}{b}$ can be rewritten as $\frac{a}{b} = q + \frac{r}{b}$ , where $q$ represents the quotient and $r$ represents the remainder.

6.

Rewrite each fraction as an equivalent mixed number.

a.

$\frac{37}{5} =$

b.

$\frac{150}{12} =$

Rational expressions work the very same way. If the degree of the numerator is greater than the degree of the denominator, the numerator can be divided by the denominator and the remainder is written as a fraction. In mathematical terms, we would say:

$\frac{a (x)}{b (x)} = q (x) + \frac{r (x)}{b (x)}$ where $q (x)$ represents the quotient and $r (x)$ represents the remainder. This is the same thing we did when we wrote equivalent multiplication statements when dividing polynomials, only in this case we start with the problem written as a fraction, not with long division.

Rewrite each expression in an equivalent form by dividing the numerator by the denominator and writing the remainder as a fraction.

7.

$\frac{x^{2} + 5 x + 7}{x + 2}$

8.

$\frac{x^{2} + 2 x + 5}{x + 3}$

In the previous lesson, when we looked at the graphs of rational functions, we did not consider the case when the numerator of the fraction is greater than the denominator. So, let’s take a closer look at the rational function from problem 7.

9.

Let $f (x) = \frac{x^{2} + 5 x + 7}{x + 2}$ . Where do you expect the vertical asymptote and the intercepts to be?

10.

Use technology to graph the function from problem 9. Relate the graph of the function to the equivalent expression that you wrote. What do you notice about the end behavior?

11.

Let’s try the same thing with problem 8. Let $f (x) = \frac{x^{2} + 2 x + 5}{x + 3}$ . Find the vertical asymptote, the intercepts, and then relate the end behavior of the graph to the equivalent expression for $f (x)$ .

12.

Using the examples from problems 10 and 11, write a process for predicting the graphs of rational functions when the degree of the numerator is one degree greater than the degree of the denominator.

Ready for More?

How would you describe the end behavior of the rational function $f (x) = \frac{x^{3} + 3 x^{2} + 2 x + 1}{x - 1}$ ?

Takeaways

Writing equivalent forms for rational expressions:

When a rational function has a common factor in the numerator and the denominator:

When the degree of the numerator is one degree greater than the degree of the denominator:

Vocabulary

slant asymptote
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned that equivalent expressions can be found for rational expressions like rational numbers when there are common factors in the numerator and denominator. When the degree of the numerator is greater than the degree of the denominator, we learned that a rational expression can be written in an equivalent form by dividing the numerator by the denominator. When this operation is performed on a rational function, the quotient indicates the end behavior or slant asymptote of the function.

Retrieval

1.

Find the roots and the domain of $f (x)$ and $g (x)$ .

a.

$f (x) = (x + 9) (x - 4) (x + 2)$

b.

$g (x) = \frac{1}{(x + 9) (x - 4) (x + 2)}$

2.

Find the domain of $h (x)$ . Then write the equation(s) of the vertical asymptote(s). $h (x) = \frac{(x + 5)}{(x + 6) (x - 3)}$