# 5.2 Power functions and polynomial functions  (Page 5/19)

 Page 5 / 19

Given the function $\text{\hspace{0.17em}}f\left(x\right)=0.2\left(x-2\right)\left(x+1\right)\left(x-5\right),\text{\hspace{0.17em}}$ express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

The leading term is $\text{\hspace{0.17em}}0.2{x}^{3},\text{\hspace{0.17em}}$ so it is a degree 3 polynomial. 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; 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.

## Identifying local behavior of polynomial functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the y- intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y- intercept $\text{\hspace{0.17em}}\left(0,{a}_{0}\right).\text{\hspace{0.17em}}$ The x- intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x- intercept. See [link] .

## Intercepts and turning points of polynomial functions

A turning point    of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y- intercept is the point at which the function has an input value of zero. The x -intercepts are the points at which the output value is zero.

Given a polynomial function, determine the intercepts.

1. Determine the y- intercept by setting $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ and finding the corresponding output value.
2. Determine the x -intercepts by solving for the input values that yield an output value of zero.

## Determining the intercepts of a polynomial function

Given the polynomial function $\text{\hspace{0.17em}}f\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-4\right),\text{\hspace{0.17em}}$ written in factored form for your convenience, determine the y - and x -intercepts.

The y- intercept occurs when the input is zero so substitute 0 for $\text{\hspace{0.17em}}x.$

$\begin{array}{ccc}\hfill f\left(0\right)& =& {\left(0\right)}^{4}-4{\left(0\right)}^{2}-45\hfill \\ & =& -45\hfill \end{array}$

The y- intercept is (0, 8).

The x -intercepts occur when the output is zero.

$0=\left(x-2\right)\left(x+1\right)\left(x-4\right)$
$\begin{array}{ccccccccccc}\hfill x-2& =& 0\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x+1& =& 0\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x-4& =& 0\hfill \\ \hfill x& =& 2\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x& =& -1\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x& =& 4\hfill \end{array}$

The x -intercepts are $\text{\hspace{0.17em}}\left(2,0\right),\left(–1,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,0\right).$

We can see these intercepts on the graph of the function shown in [link] .

## Determining the intercepts of a polynomial function with factoring

Given the polynomial function $\text{\hspace{0.17em}}f\left(x\right)={x}^{4}-4{x}^{2}-45,\text{\hspace{0.17em}}$ determine the y - and x -intercepts.

The y- intercept occurs when the input is zero.

$\begin{array}{ccc}\hfill f\left(0\right)& =& {\left(0\right)}^{4}-4{\left(0\right)}^{2}-45\hfill \\ & =& -45\hfill \end{array}$

The y- intercept is $\text{\hspace{0.17em}}\left(0,-45\right).$

The x -intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

$\begin{array}{ccc}\hfill f\left(x\right)& =& {x}^{4}-4{x}^{2}-45\hfill \\ & =& \left({x}^{2}-9\right)\left({x}^{2}+5\right)\hfill \\ & =& \left(x-3\right)\left(x+3\right)\left({x}^{2}+5\right)\hfill \end{array}$
$\phantom{\rule{2em}{0ex}}0=\left(x-3\right)\left(x+3\right)\left({x}^{2}+5\right)$

The x -intercepts are $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(–3,0\right).$

We can see these intercepts on the graph of the function shown in [link] . We can see that the function is even because $\text{\hspace{0.17em}}f\left(x\right)=f\left(-x\right).$

Given the polynomial function $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{3}-6{x}^{2}-20x,\text{\hspace{0.17em}}$ determine the y - and x -intercepts.

y -intercept $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ x -intercepts $\text{\hspace{0.17em}}\left(0,0\right),\left(–2,0\right),\text{\hspace{0.17em}}$ and $\left(5,0\right)$

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given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
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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
Y
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master
Soo sorry (5±Root11* i)/3
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Mukhtar
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y/y+10
Mr
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Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
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
Miranda
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Jhovie
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A banana.
Yaona
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nothing up todat yet
Miranda
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Miranda Drice
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Miranda
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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
state and prove Cayley hamilton therom
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
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Propessor
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algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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