Use technology to find the maximum and minimum values on the interval
$\text{\hspace{0.17em}}[\mathrm{-1},4]\text{\hspace{0.17em}}$ of the function
$\text{\hspace{0.17em}}f(x)=-0.2{(x-2)}^{3}{(x+1)}^{2}(x-4).$
The minimum occurs at approximately the point
$\text{\hspace{0.17em}}(0,-6.5),\text{\hspace{0.17em}}$ and the maximum occurs at approximately the point
$\text{\hspace{0.17em}}(3.5,7).$
Polynomial functions of degree 2 or more are smooth, continuous functions. See
[link] .
To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See
[link],[link], and
[link] .
Another way to find the
$\text{\hspace{0.17em}}x\text{-}$ intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the
$\text{\hspace{0.17em}}x\text{-}$ axis. See
[link].
The multiplicity of a zero determines how the graph behaves at the
$\text{\hspace{0.17em}}x\text{-}$ intercepts. See
[link].
The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
The end behavior of a polynomial function depends on the leading term.
The graph of a polynomial function changes direction at its turning points.
A polynomial function of degree
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ has at most
$\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See
[link].
To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most
$\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See
[link] and
[link].
Graphing a polynomial function helps to estimate local and global extremas. See
[link].
The Intermediate Value Theorem tells us that if
$\text{\hspace{0.17em}}f(a)\text{and}f(b)\text{\hspace{0.17em}}$ have opposite signs, then there exists at least one value
$\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ between
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ for which
$\text{\hspace{0.17em}}f\left(c\right)=0.\text{\hspace{0.17em}}$ See
[link].
Section exercises
Verbal
What is the difference between an
$\text{\hspace{0.17em}}x\text{-}$ intercept and a zero of a polynomial function
$\text{\hspace{0.17em}}f?\text{\hspace{0.17em}}$
The
$\text{\hspace{0.17em}}x\text{-}$ intercept is where the graph of the function crosses the
$\text{\hspace{0.17em}}x\text{-}$ axis, and the zero of the function is the input value for which
$\text{\hspace{0.17em}}f(x)=0.$
If a polynomial function of degree
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ has
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ distinct zeros, what do you know about the graph of the function?
Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.
If we evaluate the function at
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and at
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and the sign of the function value changes, then we know a zero exists between
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b.$
If the graph of a polynomial just touches the
$\text{\hspace{0.17em}}x\text{-}$ axis and then changes direction, what can we conclude about the factored form of the polynomial?
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Someone should please solve it for me
Add 2over ×+3 +y-4 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 (3y-2)/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
graph the following linear equation using intercepts method.
2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
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=3x-2
UriEl
y=mx+b
you were already given the 'm' and 'b'.
so..
y=3x-2
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 y-axis. 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.