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## 1. End Behavior of a Function

The **end behavior** of a function tells us what happens at the tails; what happens as the independent variable (i.e. “x”) goes to negative and positive infinity.

There are three main types:

- If the
*limit*of the function goes to infinity (either positive or negative) as x goes to infinity, the end behavior is**infinite**. - If the limit of the function goes to some finite number as x goes to infinity, the end behavior is
**finite**. - There are also cases where the limit of the function as x goes to infinity
**does not exist**; these are typically oscillating functions like the sine function.

The function below, a third degree polynomial, has infinite end behavior, as do all polynomials.

## End Behavior of a Polynomial

Though a polynomial typically has infinite end behavior, a look at the polynomial can tell you what kind of infinite end behavior it has. There are two important markers of end behavior: degree and leading coefficient.

- The degree is the additive value of the exponents for each individual term.
- The leading coefficient is the number in front of the variable with the largest exponent.

The end behavior, according to the above two markers:

- If the
**degree is even**and the**leading coefficient is positive**, the function will go to positive infinity as x goes to either positive or negative infinity. We write this as f(x) → +∞, as x → −∞ and f(x) → +∞, as x → +∞.A simple example of a function like this is f(x) = x

^{2}. The graph of this function is a simple upward pointing parabola - If the
**degree is even**and the**leading coefficient is negative**, the function will go to negative infinity as x goes to either positive or negative infinity. We write this as f(x) → −∞, as x → −∞ and f(x) →−∞, as x → +∞.An example of this type of function would be f(x) = -x

^{2}; the graph of this function is a downward pointing parabola. - If the
**degree is odd**and the**leading coefficient is positive**, the function will go to positive infinity as x goes to positive infinity, and negative infinity as x goes to negative infinity. We write this as f(x) → +∞, as x→ +∞ and f(x) → −∞, as x→ −∞.An example of this is f(x) = x

^{3} - If the
**degree is odd**and the**leading coefficient is negative**, the function will go to negative infinity as x goes to positive infinity, and negative infinity as x goes to positive infinity. We write this as f(x) → +∞, as x→−∞ and f(x) → −∞, as x → +∞.An example of this is f(x) = -x

^{3}.

## 2. Local Behavior

In addition to end behavior, where we are interested in what happens at the tail end of function, we are also interested in **local behavior**, or what occurs in the *middle *of a function.

The point is to find locations where the behavior of a graph changes. These *turning points* are places where the function values switch directions. For example, a function might change from increasing to decreasing.

On the graph below there are three turning points labeled *a*, *b* and *c*:

You would typically look at local behavior when working with polynomial functions.

## 3. Example—Finding the Number of Turning Points and Intercepts

Once you know the degree, you can find the number of turning points by subtracting 1. So, where the degree is equal to N, the number of turning points can be found using N-1.

**Example question:** How many turning points and intercepts does the graph of the following polynomial function have?

f(x) = x^{3} – 4x^{2} + x + 1

Step 1: Find the number of degrees of the polynomial.

The degree in the above example is 3, since it is the highest exponent. Therefore, the function will have 3 x-intercepts.

Step 2: Subtract one from the degree you found in Step 1:

N – 1 = 3 – 1 = 2.

This function has two turning points.

## References

Math 175 5-1a Notes and Learning Goals

Retrieved from https://math.boisestate.edu/~jaimos/classes/m175-45-summer2014/notes/notes5-1a.pdf on October 15, 2018.

Wilson, J. Introduction to End Behavior. EMAT 6680. Retrieved from http://jwilson.coe.uga.edu/EMAT6680Fa06/Fox/Instructional%20Unit%20Folder/Introduction%20to%20End%20Behavior.htm on October 15, 2018.

**CITE THIS AS:**

**Stephanie Glen**. "End Behavior, Local Behavior & Turning Points" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/end-behavior/

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