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Key equations

general form of a polynomial function f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0

Key concepts

  • A power function is a variable base raised to a number power. See [link] .
  • The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
  • The end behavior depends on whether the power is even or odd. See [link] and [link] .
  • A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See [link] .
  • The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See [link] .
  • The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See [link] and [link] .
  • A polynomial of degree n will have at most n x- intercepts and at most n 1 turning points. See [link] , [link] , [link] , [link] , and [link] .

Section exercises

Verbal

Explain the difference between the coefficient of a power function and its degree.

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

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If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

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In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

As x decreases without bound, so does f ( x ) . As x increases without bound, so does f ( x ) .

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What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

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What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x , f ( x ) and as x , f ( x ) .

The polynomial function is of even degree and leading coefficient is negative.

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Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

f ( x ) = ( x 2 ) 3

Power function

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f ( x ) = x 2 x 2 1

Neither

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

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f ( x ) = 3 x + 1

Neither

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For the following exercises, find the degree and leading coefficient for the given polynomial.

7 2 x 2

Degree = 2, Coefficient = –2

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2 x 2   3 x 5 +   x 6  

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x ( 4 x 2 ) ( 2 x + 1 )

Degree =4, Coefficient = –2

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For the following exercises, determine the end behavior of the functions.

f ( x ) = x 4

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

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f ( x ) = x 4

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

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f ( x ) = 2 x 4   3 x 2 +   x 1  

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

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f ( x ) = 3 x 2 +   x 2

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

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

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For the following exercises, find the intercepts of the functions.

Questions & Answers

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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