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For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

Graph of an odd-degree polynomial.

Yes. Number of turning points is 2. Least possible degree is 3.

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Graph of an even-degree polynomial.

Yes. Number of turning points is 1. Least possible degree is 2.

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Graph of an odd-degree polynomial.

Yes. Number of turning points is 0. Least possible degree is 1.

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Graph of an odd-degree polynomial.

Yes. Number of turning points is 0. Least possible degree is 1.

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Numeric

For the following exercises, make a table to confirm the end behavior of the function.

f ( x ) = x 4 5 x 2

x f ( x )
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

as x , f ( x ) , as x , f ( x )

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f ( x ) = x 2 ( 1 x ) 2

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f ( x ) = ( x 1 ) ( x 2 ) ( 3 x )

x f ( x )
10 –504
100 –941,094
–10 1,716
–100 1,061,106

as x , f ( x ) , as x , f ( x )

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f ( x ) = x 5 10 x 4

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Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

f ( x ) = x 3 ( x 2 )

Graph of f(x)=x^3(x-2).

The y - intercept is ( 0 ,   0 ) . The x - intercepts are ( 0 ,   0 ) ,   ( 2 ,   0 ) . As x , f ( x ) , as x , f ( x )

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f ( x ) = x ( x 3 ) ( x + 3 )

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f ( x ) = x ( 14 2 x ) ( 10 2 x )

Graph of f(x)=x(14-2x)(10-2x).

The y - intercept is ( 0 , 0 ) . The x - intercepts are ( 0 ,   0 ) ,   ( 5 ,   0 ) ,   ( 7 ,   0 ) . As x , f ( x ) , as x , f ( x )

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f ( x ) = x ( 14 2 x ) ( 10 2 x ) 2

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f ( x ) = x 3 16 x

The y - intercept is ( 0 ,   0 ) . The x - intercept is ( 4 ,   0 ) ,   ( 0 ,   0 ) ,   ( 4 ,   0 ) . A s x , f ( x ) , as x , f ( x )

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f ( x ) = x 4 81

Graph of f(x)=x^3-27.

The y - intercept is ( 0 ,   81 ) . The x - intercept are ( 3 ,   0 ) ,   ( 3 ,   0 ) . As x , f ( x ) , as x , f ( x )

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f ( x ) = x 3 + x 2 + 2 x

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f ( x ) = x 3 2 x 2 15 x

Graph of f(x)=-x^3+x^2+2x.

The y - intercept is ( 0 ,   0 ) . The x - intercepts are ( 3 ,   0 ) ,   ( 0 ,   0 ) ,   ( 5 ,   0 ) . As x , f ( x ) , as x , f ( x )

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f ( x ) = x 3 0.01 x

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Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The y - intercept is ( 0 , 4 ) . The x - intercepts are ( 2 , 0 ) , ( 2 , 0 ) . Degree is 2.

End behavior: as x , f ( x ) , as x , f ( x ) .

f ( x ) = x 2 4

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The y - intercept is ( 0 , 9 ) . The x - intercepts are ( 3 , 0 ) , ( 3 , 0 ) . Degree is 2.

End behavior: as x , f ( x ) , as x , f ( x ) .

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The y - intercept is ( 0 , 0 ) . The x - intercepts are ( 0 , 0 ) , ( 2 , 0 ) . Degree is 3.

End behavior: as x , f ( x ) , as x , f ( x ) .

f ( x ) = x 3 4 x 2 + 4 x

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The y - intercept is ( 0 , 1 ) . The x - intercept is ( 1 , 0 ) . Degree is 3.

End behavior: as x , f ( x ) , as x , f ( x ) .

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The y - intercept is ( 0 , 1 ) . There is no x - intercept. Degree is 4.

End behavior: as x , f ( x ) , as x , f ( x ) .

f ( x ) = x 4 + 1

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Real-world applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of d , the number of days elapsed.

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A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of m , the number of minutes elapsed.

V ( m ) = 8 m 3 + 36 m 2 + 54 m + 27

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A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x inches and the width increased by twice that amount, express the area of the rectangle as a function of x .

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An open box is to be constructed by cutting out square corners of x - inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x .

V ( x ) = 4 x 3 32 x 2 + 64 x

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A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ( x ).

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Questions & Answers

if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
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Kanniyappan Reply
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Rightspect
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chesky
Give me algebra questions
Rightspect
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Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-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
Adityasuman x= - 1
Aditya
y=x+1
gary
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
Shadow Reply
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
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
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
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
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
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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