Lesson 7 What’s the End Game? Develop 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 $x \to - \infty, f (x) \to \underset{―}{}$ and as $x \to \infty, f (x) \to \underset{―}{}$

3.