Lesson 3 Rational Thinking Solidify Understanding

Jump Start

Graph each function:

1.

$y = - 2 + \frac{1}{x + 3}$

2.

$y = 1 - \frac{1}{x - 2}$

Learning Focus

Define a rational function.

Explore rational functions, and find patterns that predict the asymptotes and intercepts.

What is a rational function? What does the graph of a rational function look like?

Open Up the Math: Launch, Explore, Discuss

The broad category of functions that contains the function $f (x) = \frac{1}{x}$ is called rational functions. A rational number is a ratio of integers. A rational function is a ratio of polynomials. Since polynomials come in many forms, constant, linear, quadratic, cubic, etc., we can expect rational functions to come in many forms, too. Some examples are:

$f (x) = \frac{1}{x}$	$f (x) = \frac{x + 1}{x - 3}$	$f (x) = \frac{x - 4}{x^{3} + 2 x^{2} - 4 x + 1}$	$f (x) = \frac{x^{2} - 4}{(x - 3) (x + 1)}$
Degree of the numerator $=$ $0$ Degree of the denominator $=$ $1$	Degree of the numerator $=$ $1$ Degree of the denominator $=$ $1$	Degree of the numerator $=$ $1$ Degree of the denominator $=$ $3$	Degree of the numerator $=$ $2$ Degree of the denominator $=$ $2$

$f (x) = \frac{1}{x}$

$f (x) = \frac{x + 1}{x - 3}$

$f (x) = \frac{x - 4}{x^{3} + 2 x^{2} - 4 x + 1}$

$f (x) = \frac{x^{2} - 4}{(x - 3) (x + 1)}$

Degree of the numerator $=$ $0$

Degree of the denominator $=$ $1$

Degree of the numerator $=$ $1$

Degree of the denominator $=$ $1$

Degree of the numerator $=$ $1$

Degree of the denominator $=$ $3$

Degree of the numerator $=$ $2$

Degree of the denominator $=$ $2$

In today’s task, you are going to look for patterns in the following forms so that you can complete the chart:

	How to find the vertical asymptote:	How to find the horizontal asymptote:	How to find the intercepts:
Degree of the numerator $<$ Degree of the denominator
Degree of the numerator $=$ Degree of the denominator

You are given several different rational functions. Start by identifying the degree of the numerator and denominator and using technology to graph the function. As you are working, look for patterns that will help you complete the table. You need to find a quick way to identify the horizontal and vertical asymptotes when you see the equation of a rational function, as well as noticing other patterns that will help you analyze and graph the function quickly. The last two graphs are there so you can experiment with your own rational functions and test your theories.

1.

$y = \frac{x + 3}{x - 2}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

2.

$y = \frac{x}{(x + 2) (x - 3)}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

3.

$y = \frac{1}{{(x + 4)}^{2}}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

4.

$y = \frac{3 x - 1}{x + 4}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

5.

$y = \frac{x^{2} - 3 x}{(x - 2) (x + 1) (x + 3)}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

6.

$y = \frac{2 x^{2} + 3}{(2 x + 5) (2 x - 5)}$

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

7.

Your own rational function:

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

8.

Your own rational function:

Degree of Numerator:

Degree of Denominator:

Horizontal Asymptote:

Vertical Asymptote:

Intercepts:

Ready for More?

Compare the graphs of the functions $y = \frac{3 x + 5}{x + 2}$ and $y = 3 - \frac{1}{x + 2}$ and explain any similarities or differences that you find.

Takeaways

Our final conclusions:

How to Find the Vertical Asymptote:

How to Find the Horizontal Asymptote:

How to Find the Intercepts:

Degree of the Numerator $<$ Degree of the Denominator

$x$ -intercepts:

$y$ -intercept:

Degree of the Numerator $=$ Degree of the Denominator

Vocabulary

rational function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned to identify the horizontal and vertical asymptotes of a rational function by comparing the degree of the numerator to the degree of the denominator. The vertical asymptotes occur where the function is undefined, and the horizontal asymptote describes the end behavior of the function. Finding the intercepts is the same as other functions we know but there are ways to be more efficient with rational functions.

Retrieval

1.

Add $\frac{5}{21} + \frac{9}{14}$ . Explain each step you need to take to add the two fractions.

2.

Rewrite $\frac{24}{32}$ . Explain the process and justify each step.