# 5.2 Power functions and polynomial functions  (Page 2/19)

 Page 2 / 19

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol $\text{\hspace{0.17em}}\infty \text{\hspace{0.17em}}$ for positive infinity and $\text{\hspace{0.17em}}\mathrm{-\infty }\text{\hspace{0.17em}}$ for negative infinity. When we say that “ $x\text{\hspace{0.17em}}$ approaches infinity,” which can be symbolically written as $\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}$ we are describing a behavior; we are saying that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is increasing without bound.

With the positive even-power function, 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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches positive or negative infinity, the $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ values increase without bound. In symbolic form, we could write

[link] shows the graphs of $\text{\hspace{0.17em}}f\left(x\right)={x}^{3},\text{\hspace{0.17em}}g\left(x\right)={x}^{5},$ and $\text{\hspace{0.17em}}h\left(x\right)={x}^{7},$ which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, 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 $\text{\hspace{0.17em}}f\left(x\right)={x}^{n}\text{\hspace{0.17em}}$ reveal symmetry of one kind or another. First, in [link] we see that even functions of the form even, are symmetric about the $\text{\hspace{0.17em}}y\text{-}$ axis. In [link] we see that odd functions of the form  odd, are symmetric about the origin.

For these odd power functions, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ decreases without bound. As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches positive infinity, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ increases without bound. In symbolic form we write

The behavior of the graph of a function as the input values get very small ( $\text{\hspace{0.17em}}x\to -\infty \text{\hspace{0.17em}}$ ) and get very large ( $\text{\hspace{0.17em}}x\to \infty \text{\hspace{0.17em}}$ ) is referred to as the end behavior    of the function. We can use words or symbols to describe end behavior.

[link] shows the end behavior of power functions in the form $\text{\hspace{0.17em}}f\left(x\right)=k{x}^{n}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is a non-negative integer depending on the power and the constant.

Given a power function $\text{\hspace{0.17em}}f\left(x\right)=k{x}^{n}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is a non-negative integer, identify the end behavior.

1. Determine whether the power is even or odd.
2. Determine whether the constant is positive or negative.
3. Use [link] to identify the end behavior.

## Identifying the end behavior of a power function

Describe the end behavior of the graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{8}.$

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches infinity, the output (value of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ ) increases without bound. We write as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty .\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, the output increases without bound. In symbolic form, as We can graphically represent the function as shown in [link] .

## Identifying the end behavior of a power function.

Describe the end behavior of the graph of $\text{\hspace{0.17em}}f\left(x\right)=-{x}^{9}.$

The exponent of the power function is 9 (an odd number). Because the coefficient is $\text{\hspace{0.17em}}–1\text{\hspace{0.17em}}$ (negative), the graph is the reflection about the $\text{\hspace{0.17em}}x\text{-}$ axis of the graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{9}.\text{\hspace{0.17em}}$ [link] shows that as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches infinity, the output decreases without bound. As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, the output increases without bound. In symbolic form, we would write

can you solve it step b step
what is linear equation with one unknown 2x+5=3
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2