



Key equations
general form of a polynomial function 
$$f(x)={a}_{n}{x}^{n}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$$ 
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
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have at most
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$
x intercepts and at most
$\text{\hspace{0.17em}}n1\text{\hspace{0.17em}}$ 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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In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).$
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What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty \text{\hspace{0.17em}}$ and as
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty .\text{\hspace{0.17em}}$
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.
For the following exercises, find the degree and leading coefficient for the given polynomial.
For the following exercises, determine the end behavior of the functions.
$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f(x)=2{x}^{4}3{x}^{2}+x1$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f(x)={x}^{2}(2{x}^{3}x+1)$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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For the following exercises, find the intercepts of the functions.
Questions & Answers
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x4/x^22x15 then
x is different from 5&3
Seid
how to prroved cos⁴xsin⁴x= cos²xsin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴xsin⁴x = cos²xsin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me
Add 2over ×+3 +y4 over 5
simplify (×+a)with square root of two ×root 2 all over a
multiply 1over ×y{(×y)(×+y)} over ×y
For the first question, I got (3y2)/15
Second one, I got Root 2
Third one, I got 1/(y to the fourth power)
I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=2
graph the following linear equation using intercepts method.
2x+y=4
Ashley
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x2
UriEl
y=mx+b
you were already given the 'm' and 'b'.
so..
y=3x2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line.
where m = the slope & b = where the line crosses the yaxis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
0=3x2
2=3x
x=3/2
then .
y=3/2X2
I think
Given
co ordinates for x
x=0,(2,0)
x=1,(1,1)
x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=b+_Гb2(4ac)
______________
2a
I've run into this:
x = r*cos(angle1 + angle2)
Which expands to:
x = r(cos(angle1)*cos(angle2)  sin(angle1)*sin(angle2))
The r value confuses me here, because distributing it makes:
(r*cos(angle2))(cos(angle1)  (r*sin(angle2))(sin(angle1))
How does this make sense? Why does the r distribute once
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis
vice versa with y=x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As
'f(x)=y'.
According to Google,
"The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :)
Thomas
GREAT ANSWER THOUGH!!!
Darius
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks.
"Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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