WOOHHH. Mr. Tarrou. Let's this polynomial graphing function thing that we just started
to talk about. Ok, graphs of polynomial functions. We just got done talking about what a polynomial
is, what a polynomial graph looks like, and we started talking about the end behavior
of a graph looks like. Now I want to talk about where these functions are going to cross
the x axis and how they are...actually not necessarily cross the x axis but touch the
x axis. Something called multiplicity of the zeros. Here we have again our little parent
function which is a parabola. That parabola will touch the origin at (0,0) and go back
up again because my leading coefficient is positive and my degree is even, so up on both
sides. What is this going to look like? Well, we are talking about x intercepts so before
we go and just finish looking at this picture. Let's talk about finding the x intercept.
When a graph crosses/touches the x axis the y value is zero. So what we are finding here
when we find x intercepts, solutions, roots, zeros, whatever you want to call it...they
are x intercepts. That is when we let y equal zero. So this is going to be zero equals x
squared. We are going to find the x intercept of this little bad boy here so we will let
y equal zero. zero equals x squared minus x minus 6. Let's find the solutions, let's
find the x intercepts. Well clearly this is x times x and that means that my x intercept
is going to be zero. Here this is factorable. What are the factors of six that subtract
to give you the middle term? That is going to be x minus three times x plus 2. Now that
this is factored I can solve for... that was 2. I can go ahead and solve this by setting
each of these factors equal to zero and get x equals three and x equals negative two.
So why is it, and I already know the answer, but why is it that this graph touches the
x value of zero and if you type into your calculator this graph is going to go through
negative two and through the value of three and look like this. Why is one graph touching
the x axis at zero and this equation has the graph going through the values of negative
two and three. The reason why is because of something called multiplicity. This x of this
x intercept of zero has an even multiplicity. When a zero, a solution, root, zero, x intercept
has an exponent which is even or shows up two, four, six, or whatever times in your
solution that means that it touches the x axis. Here this three came from a factor that
had an exponent that was odd, the two... the negative two came from a factor that has an
odd exponent. I know i just misspoke there for a second. But the three came from here
that has an odd power, the negative two came from this factor that had an odd power, and
when you have an odd multiplicity then the graph goes through those points. BAM!!! New
concept... Multiplicity. Even power touches, odd power...zeros...goes through those values.
So we have talked about end behavior. Now we know how to find x intercepts and we know
if it is going to touch that point or go through that point. Even power on the zero touches,
odd power on my zeros means the graph goes through. Now we are going to talk about y
intercepts. Y intercepts are always pretty simple. All you have to do for a y intercept
is let x equal zero. So if I wanted the y intercept of this second very simple example,
this parabola, I just go ok... F of zero equals zero squared minus zero minus six. F of zero
equals negative six or that is my intercept. So if I wanted to now come through and mark
my y axis at negative six, I would have a better sketch of what this parabola looks
like. Now you might be going why are we doing all this work for a parabola and why aren't
we talking about the bottom of the curve? Why are we not talking about where the vertex
is? Well, if you want to graph polynomials you are going to be doing a lot more complicated
stuff than just parabolas. So finding the bottom of those dips or the tops, or the relative
maximums or relative minimums, that is beyond the scope of Precalculus that we are going
to be taking a look at. You will need a graphing calculator for that and some possible some
Calculus techniques. So we are going to just be talking about the x and y intercepts and
the end behavior. Now one thing before I go and start erasing blindly is another thing
that you might want to think about trying to sketch a polynomial function is if it is
either even or odd. Because if a function is even, and I have spoken a lot about this
in other lessons, that is when f(-x) is equal to f(x). You plug in opposite numbers and
you get the same answer like a parabola centered on the y axis. This parabola is even. Negative
x and positive x yields the same y value. That symmetry to the y axis would be very helpful in you drawing
a graph or a sketch of polynomial function. On the left and right hand side of the y axis
you are going to see exactly the same thing. So might want to check for even symmetry before
you move on to see if you can get balance on the left and right hand side of the y axis.
You also have the possibility of the function being odd. If a function is odd, that means
that you plug in opposite x's and find opposite y's. Odd functions have 180 degree rotational
symmetry. So like your cubic function that I just had up here a second ago. If I can
take the board and turn it over 180 degrees, then I would see exactly the same picture.
That might also help you sketch the graph of a polynomial if you know that it was odd.
So even and odd functions, that is a nice little thing to check for to speed up or make
sure that your sketches are a little bit more accurate. Well that is me talking about a
lot of concepts and ideas but no examples. Let me see if I can get two examples done
before my time runs out. And send you on your merry way to do some homework and graph some
polynomial functions. So here is one. F of x is four times x plus two times x minus one
squared. Now this is already in factored form so I am going to try and do this as quickly
as I can. This x is going to be raised to an even power, so we are talking about a polynomial
here that has a second degree. Maybe I will just show that real quick. x squared minus
two x plus one. Then this trinomial would get multiplied to this one if we were to actually
multiply all this out and this second degree would match up with this first degree and
come up with a third degree and then your leading coefficient would be positive. Now
I want to show more than one example so I am not going to distribute all this out. But
if you were to, your leading coefficient would be 4 x to the third power with a lot of other
stuff going on. So this polynomial is a third degree and has a positive leading coefficient.
So because of that the end behavior is going to be down to the left and up to the right
because of the positive leading coefficient and the exponent of three. That is the end
behavior. Now let's talk about the zeros...the x intercepts. Well, why do we factor equations.
We factor equations to help us solve for zero, to help us solve for solutions, roots, zeros.
This is already factored so basically if x plus two is equal to zero, which would be
when x equals negative two...of if this x minus one is equal to zero which would be
x equals one this whole thing would be zero and those are our solutions. This is in factored
form, set each factor equal to zero, and find your solutions. In this case my solutions
are going to be x intercepts. Now what is the graph going to do at negative two? Well,
my degree is odd so it is going to go through negative two. This zero of one had an even
power, the graph is going to touch the x value of one because my degree is even. Even touch,
odd power go through. Now I know my graph is going to go through the x value of negative
two so it is going to look something like this and it is going to touch the x axis at
one, so my graph is going to come up do something around the y axis and then it is going to
come down and touch the x value of one and continue up to the right somehow. I don't
know how tall that peak is going to be, but I can find the y intercept. You find the y
intercept by simply letting x equal zero. So, for my y intercept I let x equal zero
and it will be four times zero plus two times zero minus one squared. That is going to be
four times two times one which is eight. So i know that my graph is going to going to
go up to a y value of 8...1,2,3,4,5,6,7,8 and go through here somehow, go down to one,
and then boom up to the right. Now you can take your graphing calculator and go ahead
and put this equation in and see how close I am. Now I did not check for even and odd
to see if there is any symmetry, but I want to do two examples before the bell rings..or
my time runs out. This a third degree which means that you have up to n minus one bends,
so my degree of three has two bends. And I have an x intercept of negative two and an
x intercept of one...but is only touching here because remember the exponent, the power,
the multiplicity is even. ALRIGHT...BAM!!! Let's try one more. This one will not be factored.
What is this sketch going to look like? Again, odd and positive. Odd and positive...down
to the left and up to the right. So again down to the left and up to the right because
of the odd degree and the positive leading coefficient. Let's talk about the y intercept.
The y intercept, this in the reverse order of the previous question, I will let x equal
zero and that is zero, zero, zero, if I plug zero in for the x it is 0,0,0, ok my y intercept
is negative 12. So down here at negative twelve the graph is going to go through the y axis.
Let's talk about the x intercepts. I have one minute left. For the x intercept you let
y equal zero. So that is 0 equals x cubed plus three x squared minus four x minus 12.
When you have four terms the only way you could possibly factor is if it is by grouping.
The first two can give up an x squared. The last two terms can give up a negative four.
If you do your factoring by grouping correctly and it is a factorable polynomial, you will
see the same factor show up twice. You can factor that out again. OH... A difference
of squares, that is factorable again. It is going to be x plus three times x plus two
times x minus two. Sorry I am rushing. All of these have powers of one which is odd so
the graph is going to go through x equals negative three, negative two, and two. So
negative three up through that, negative two back down through that value, and then back
up through...oops...up through the value of two. So that graph is going to look roughly
like this. I hope that helpfully that helps. If it was too fast or if you want another
example let me know. I will do another one for you...BAM!!!