# 5.2 Power functions and polynomial functions  (Page 6/19)

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## Comparing smooth and continuous graphs

The degree of a polynomial function helps us to determine the number of x -intercepts and the number of turning points. A polynomial function of $\text{\hspace{0.17em}}n\text{th}\text{\hspace{0.17em}}$ degree is the product of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ factors, so it will have at most $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ roots or zeros, or x -intercepts. The graph of the polynomial function of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ must have at most $\text{\hspace{0.17em}}n–1\text{\hspace{0.17em}}$ turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function    has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve    is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

## Intercepts and turning points of polynomials

A polynomial of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have, at most, $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ x -intercepts and $\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points.

## Determining the number of intercepts and turning points of a polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x -intercepts and turning points for $\text{\hspace{0.17em}}f\left(x\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}.$

The polynomial has a degree of $\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}$ so there are at most 10 x -intercepts and at most 9 turning points.

Without graphing the function, determine the maximum number of x -intercepts and turning points for $\text{\hspace{0.17em}}f\left(x\right)=108-13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}.$

There are at most 12 $\text{\hspace{0.17em}}x\text{-}$ intercepts and at most 11 turning points.

## Drawing conclusions about a polynomial function from the graph

What can we conclude about the polynomial represented by the graph shown in [link] based on its intercepts and turning points?

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See [link] .

The graph has 2 x -intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

What can we conclude about the polynomial represented by the graph shown in [link] based on its intercepts and turning points?

The end behavior indicates an odd-degree polynomial function; there are 3 $\text{\hspace{0.17em}}x\text{-}$ intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

## Drawing conclusions about a polynomial function from the factors

Given the function $\text{\hspace{0.17em}}f\left(x\right)=-4x\left(x+3\right)\left(x-4\right),\text{\hspace{0.17em}}$ determine the local behavior.

The y -intercept is found by evaluating $\text{\hspace{0.17em}}f\left(0\right).$

$\begin{array}{ccc}\hfill f\left(0\right)& =& -4\left(0\right)\left(0+3\right)\left(0-4\\ & =& 0\hfill \end{array}$

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

The x -intercepts are found by determining the zeros of the function.

$0=-4x\left(x+3\right)\left(x-4\right)$
$\begin{array}{ccccccccccc}\hfill x& =& 0\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x+3& =& 0\hfill & \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& \hfill x-4& =& 0\hfill \\ x& =& 0& \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& x& =& -3& \phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}& x& =& 4\end{array}$

The x -intercepts are $\text{\hspace{0.17em}}\left(0,0\right),\left(–3,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,0\right).$

The degree is 3 so the graph has at most 2 turning points.

Given the function $\text{\hspace{0.17em}}f\left(x\right)=0.2\left(x-2\right)\left(x+1\right)\left(x-5\right),\text{\hspace{0.17em}}$ determine the local behavior.

The $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{\hspace{0.17em}}\left(2,0\right),\left(-1,0\right),$ and $\text{\hspace{0.17em}}\left(5,0\right),\text{\hspace{0.17em}}$ the y- intercept is $\text{\hspace{0.17em}}\left(0,\text{2}\right),\text{\hspace{0.17em}}$ and the graph has at most 2 turning points.

Access these online resources for additional instruction and practice with power and polinomial functions.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin