Lesson 9 Is This the End? Solidify Understanding

Learning Focus

Find patterns in the end behavior of polynomial functions.

Describe the end behavior of a function using appropriate notation.

What conclusions can be drawn about the end behavior of polynomial functions?

How does the end behavior of polynomials compare to other functions we know?

Open Up the Math: Launch, Explore, Discuss

Previously, you have compared and analyzed growth rates of polynomial (mostly linear and quadratic) and exponential functions. In this task, we are going to analyze rates of change and end behavior by comparing various expressions to find patterns that we can use to predict end behavior.

Part I: Seeing patterns in end behavior

$f (x) = 2^{x}$	$p (x) = x^{3} + x^{2} - 4$	$g (x) = x^{2} - 20$
$h (x) = x^{5} - 4 x^{2} + 1$	$k (x) = x + 30$	$m (x) = x^{4} - 1$
$r (x) = x^{5}$	$n (x) = {(\frac{1}{2})}^{x}$	$q (x) = x^{6}$

1.

Using the graph provided, write the given functions vertically, from greatest to least for $x = 0$ . Put the function with the greatest value on top and the function with the smallest value on the bottom. Put functions with the same values at the same level. An example, $l (x) = x^{7}$ , has been placed on the graph to get you started.
What determines the value of a polynomial function at $x = 0$ ? Is this true for other types of functions?
Write the same expressions on the graph in order from greatest to least when $x$ represents a very large number (this number is very large, so we say that it is approaching positive infinity). If the value of the function is positive, put the function in Quadrant I. If the value of the function is negative, put the function in Quadrant IV. An example has been placed for you.
What determines the end behavior of a polynomial function for very large values of $x$ ?
Write the same functions in order from greatest to least when $x$ represents a number that is approaching negative infinity. If the value of the function is positive, place it in Quadrant II; if the value of the function is negative, place it in Quadrant III. An example is shown on the graph.
What patterns do you see in the polynomial functions for $x$ values approaching negative infinity? What patterns do you see for exponential functions? Use graphing technology to test these patterns with a few more examples of your choice.

2.

How would the end behavior of the polynomial functions change if the lead terms were changed from positive to negative?

Part II: Using end behavior patterns

For each situation:

Determine the function type. If it is a polynomial, state the degree of the polynomial and whether it is an even-degree polynomial or an odd-degree polynomial.
Describe the end behavior based on your knowledge of the function. Use the format:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

3.

a.

$f (x) = 3 + 2 x$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

b.

$f (x) = x^{4} - 16$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

c.

$f (x) = 3^{x}$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

d.

$f (x) = x^{3} + 2 x^{2} - x + 5$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

e.

$f (x) = - 2 x^{3} + 2 x^{2} - x + 5$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

f.

$f (x) = \log_{2} x$

Function type:

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

4.

Use the graphs to describe the end behavior of each function by completing the statements.

a.

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

b.

As $x \to - \infty$ , $f (x) \to$ and as $x \to \infty$ , $f (x) \to$ .

5.

How does the end behavior for quadratic functions connect with the number and type of roots for these functions? How does the end behavior for cubic functions connect with the number and type of roots for cubic functions?

Part III: Even and Odd Functions

Some functions that are not polynomials may be categorized as even functions or odd functions. When mathematicians say that a function is an even function, they mean something very specific.

6.

Let’s see if you can figure out what the definition of an even function is with these examples. In each case, compare the even function with the function that is not even on the same row and write down the difference you notice.

a.

Even function:

$f (x) = x^{2}$

Not an even function:

$g (x) = 2^{x}$

Differences:

b.

Even function:

$f (x) = x^{4} - 3$

Not an even function:

$g (x) = x (x + 3) (x - 2)$

Differences:

c.

Even function:

$f (x) = - | x | + 4$

Not an even function:

$g (x) = - | x + 4 |$

Differences:

d.

Even function:

$f (2) = 5 and f (- 2) = 5$

Not an even function:

$g (2) = 3 and g (- 2) = 5$

Differences:

7.

What do you observe about the characteristics of an even function?

8.

The algebraic definition of an even function is:

$f (x)$ is an even function if and only if $f (x) = f (- x)$ for all values of $x$ in the domain of $f$ .

What are the implications of the definition for the graph of an even function?

9.

Are all even-degree polynomials even functions? Use examples to explain your answer.

10.

Let’s try the same approach to figure out a definition for odd functions.

a.

Odd function:

$f (x) = x^{3}$

Not an odd function:

$g (x) = \log_{2} x$

Differences:

b.

Odd function:

$f (x) = - x^{5}$

Not an odd function:

$g (x) = x^{3} + 3 x - 7$

Differences:

c.

Odd function: $f (x) = \frac{1}{x}$

Not an odd function:

$g (x) = 2 x - 3$

Differences:

d.

Odd function:

$f (2) = 3 and f (- 2) = - 3$

Not an odd function:

$g (2) = 3 and g (- 2) = 5$

Differences:

11.

What do you observe about the characteristics of an odd function?

12.

The algebraic definition of an odd function is:

$f (x)$ is an odd function if and only if $f (- x) = - f (x)$ for all values of $x$ in the domain of $f$ .

Explain how each of the examples of odd functions above meet this definition.

13.

How can you tell if an odd-degree polynomial is an odd function?

14.

Are all functions either odd or even?

Ready for More?

1.

Find the roots and end behavior for $g (x) = - (x - 3) (x - 3 i) (x + 3 i)$

Roots:

End behavior:

As $x \to \infty$ , $f (x) \to$ and as $x \to - \infty$ , $f (x) \to$ .

2.

What do you predict the graph will look like?

Takeaways

At $x = 0$ :

As $x \to \infty :$

As $x \to - \infty :$

Adding Notation, Vocabulary, and Conventions

Inputs:

Outputs:

Input-Output Relationships:

Even Function:

Odd Function:

Vocabulary

Lesson Summary

In this lesson, we examined the end behavior of polynomial and exponential functions. We found patterns that allow us to predict the end behavior for polynomials. We learned to use notation to write the end behavior for functions. We learned the definition of odd and even functions and how to identify them by their symmetries.

Retrieval

1.

Multiply: $(x + 7) (x - 7)$

2.

Multiply: $(x + 5) (x^{2} - 5 x + 25)$

3.

Fill in the conjugate of the given expression and multiply.

$(5 + 8 i) ($ $) =$ .