Introduction to Power and Polynomial Functions: Learn It 2

Identifying End Behavior of Power Functions

The behavior of the graph of a function as the input values get very small ( [latex]x\to -\infty[/latex] ) and get very large ( [latex]x\to \infty[/latex] ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior. Let’s look at the graph of power functions with a non-negative integer exponent that has the form [latex]f(x) = ax^n[/latex] where [latex]n[/latex] is a non-negative integer ([latex]0, 1, 2, 3, ...[/latex]).
a graph with several curves representing functions of the form f(x) = x^n, where n is an even integer. The functions graphed are f(x) = x^2 in red, g(x) = x^4 in blue, h(x) = x^6 in green, k(x) = x^8 in orange, and p(x) = x^10 in purple. All curves are symmetrical about the y-axis (indicating that they are even functions) and share the point (0,0). As n increases, the graphs become steeper near x = ±1, and the curves flatten around the origin, making them narrower for larger even powers. The legend on the bottom-right corner shows the color coding for each function.

Even-Power Functions

The graph shows the graphs of [latex]h\left(x\right)={x}^{2},g\left(x\right)={x}^{4}[/latex], and [latex]f\left(x\right)={x}^{6}[/latex], which are all power functions with even, whole-number powers. Notice that these

graphs have similar shapes, very much like that of the quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\infty[/latex] for positive infinity and [latex]-\infty[/latex] for negative infinity. When we say that “[latex]x[/latex] approaches infinity,” which can be symbolically written as [latex]x\to \infty[/latex], we are describing a behavior; we are saying that [latex]x[/latex] is increasing without bound.

even-power functions

With even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers.

 

Equivalently, we could describe this behavior by saying that as [latex]x[/latex] approaches positive or negative infinity, the [latex]f(x)[/latex] values increase without bound.

[latex]\text{as }x\to \pm \infty , f(x) \to \infty[/latex]

a graph with several curves representing functions of the form f(x) = x^n, where n is an odd integer. The functions graphed are f(x) = x^3 in red, g(x) = x^5 in blue, h(x) = x^7 in green, k(x) = x^9 in orange, and p(x) = x^11 in purple. All curves are symmetrical about the origin (indicating that they are odd functions) and pass through the point (0,0). As n increases, the graphs become steeper near x = ±1, and the curves flatten more around the origin. The larger the odd power, the closer the curves stay to the x-axis near zero, but they grow faster as x moves further away from zero. The legend on the bottom-right corner shows the color coding for each function.

Odd-Power Functions

The graph shows [latex]f\left(x\right)={x}^{3},g\left(x\right)={x}^{5}, \text{ and } h\left(x\right)={x}^{7}[/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.

These examples illustrate that functions of the form [latex]f(x)={x}^{n}[/latex] reveal symmetry of one kind or another. First, in the even-powered power functions, we see that even functions of the form [latex]f(x)={x}^{n}\text{, }n\text{ even,}[/latex] are symmetric about the y-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ odd,}[/latex] are symmetric about the origin.

odd-power functions

For these odd power functions, as [latex]x[/latex] approaches negative infinity, [latex]f\left(x\right)[/latex] decreases without bound. As [latex]x[/latex] approaches positive infinity, [latex]f(x)[/latex] increases without bound.

 

[latex]\begin{array}{c}\text{as } x\to -\infty , f(x) \to -\infty \\ \text{as } x\to \infty , f(x) \to \infty \end{array}[/latex]

The table below shows the end behavior of power functions of the form [latex]f\left(x\right)=a{x}^{n}[/latex] where [latex]n[/latex] is a non-negative integer depending on the power and the constant.

Even power Odd power

Positive constant

a > 0

 Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to positive infinity.
Negative constant

a < 0

 Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity.

 

 

 

Describe the end behavior of the graph of [latex]f\left(x\right)={x}^{8}[/latex].