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

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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

By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
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Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
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4
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remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
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Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
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x²-2x+9-4x²+12x-20 -3x²+10x+11
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x²-2x+9-4x²+12x-20 -3x²+10x+11
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(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
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Soo sorry (5±Root11* i)/3
master
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2x²-6x+1=0
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explain and give four example of hyperbolic function
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y/y+10
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Find area common to the parabola y2 = 4ax and x2 = 4ay.
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A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
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thanks
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What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
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a function
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a function
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given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
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jai
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Miranda Drice
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Miranda
I am living in india
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what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda