We’re going to look at determining endbehavior and intercepts to graph a polynomial.
So first, we need to remindourselves about a few things.
First is the end behavior. So, when we’re lookingat end behavior, we’re talking about the power of the polynomial – so all thepowers of X together.
If the power of your polynomial is even, then your graphon the ends will behave like an X squared.
end behavior of a function.
So, if you have an even powerpolynomial and it’s a positive, your ends will both be up like a positivequadratic, and if it’s negative, your ends will both be down like a negativequadratic.
Now, we don’t know what’s happening here in the middle, that’s whatend – not end behavior, excuse me – that’s what our intercepts and ourmultiplicities are going to help us with, but it we know what’s happening on theends. Similarly, with odd powers, again this is the power of the wholepolynomial, they behave like a cubic.end behavior calculator
So positive would be down to up andnegative would be up to down. Again, we’re not a hundred percent sure what’shappening here in the middle,
that’s going to be determined by intercepts andmultiplicities, but we know what’s happening on the ends. So next let’s talkabout multiplicity.
So firs,t multiplicity refers to the power of the factor.
And let me remind you – let’s go down tothe bottom here – that a factor would be X minus 4, but that the zero related tothat factor would be X equals positive 4, right? Because you would set up X minus 4 equals 0 etc. Okay, so when we look at the power of a factor if the factor,
it hasan even power so 2 4 6 8 etc., then that means that the graph will touch at thezero meaning it’ll, I call it a bounce rather than a touch.
If the multiplicity of the factor is odd your graph will cross the axis at thatzero. Okay so even – touch or bounce, odd – cross. Okay, for the y-intercept.
end behavior of a function
For they-intercept, that happens when the X is equal to zero. So we’ll just set it set Xequal to zero and see what happens. So let’s look at one. So, first things first.Let’s talk about the end behavior.
If you look at this one, we have an X squared times an X squared, right? We’re justlooking at the power,
we’re not looking at anything else. So that would give mean X to the fourth. So this is even and we don’t have a negative sign, so it’spositive,
so my end behavior is up up. Again, we’re not we’re gonna figure outwhat’s happening there in the middle.
Okay, so that’s the first part. Next, weneed to look at the multiplicity. So this first factor, X + 1,has a multiplicity of 2, which is even, which means that the graph will touch atX = -1, so we’ll go ahead and put a little point on our graph at-1.
For the second factor of X – 2, it has a multiplicity of 2, whichis even, so my graph will touch at X = 2. Now let’s look at they intercept.
If X is 0, then y would be a positive 1 squared times a -2squared, right? I just zeroed out those two X’s, so that gives me what – oh, Ididn’t square that, sorry – so that’s going to give me a positive 4, so I’ll put apoint on my graph at positive 4. So, to graph, we know thatwe’re going to start up we touch here at our negative 1,
we’re gonna have to goup and get this 4, come down and bounce at the 2, and then go up again.
end behavior calculator
Let’s lookat another one. So, here, first thing let’s look at our end behavior. So, we need tolook at the power of the polynomial – We have a negative, then we have an X to thefirst power with the first factor,
and X to the second power with the secondfactor, and an X squared with the third factor,
so that gives me a negative X tothe fifth,.So it’s negative and it’s odd, so it’s going to behave like annegative cubic, which is up to down. And again, we’re not sure what’s going tohappen here in the middle. Next let’s look at our multiplicities.
So, for ourfirst factor here, for X – 2, it has a multiplicity of 1, so it’s odd so we’regoing to cross at X = 2.
end behavior of rational functions
The next two factors are both even, so they’reboth touches or bounces, so we’re going to touch at X = -1 andpositive 1. Let’s go ahead and put those points on here, so we have a 2, a negative1, and a positive 1. Next, we’re going to look at the y-intercept.
So we keep thenegative out front here, but in the first factor we’ll be left with a -2,the second factor will be left with a 1 squared, which is 1, and the third factorwill be left with a -1 squared,
which is 1. So we have the y intercept ispositive 2, and we’ll put a point on our graph up at 2. So, now to graph this puppy, we’re going to start up up here and we are going to touch at negative 1,
we’vegot to come up and grab this y-intercept, and we’re going to touch at a positive 1.I’m going to come up a little bit, we’re not sure how much – it’s fine, that doesn’thave to be perfect – but then at 2 we cross, so it comes down which is what wewould expect it to do given our end behavior.
Ok, let’s do it again.
So, here wehave a negative, looking at the end behavior,
in the power we have a negativeX squared and an X squared and an X. So again, that’s a negative Xto the fifth, so that’s kind of what we just did. Right?
end behavior of a function
Negative odd, so up todown. Again, that’s just the end behavior. We’re not sure what’s gonna happen inthe middle. Next, looking at our multiplicities.
For the first here wehave – Don’t forget about this guys, this is a factor, right? And that’s a zerowould be the zero, right? It’d have a zero at zero,
but the multiplicity is 2, whichmeans it’s going to touch at X = 0 or bounce. So I’ll put a point atzero.
Looking at the X + 2 squared, that’s also a touch at X = -2. And then we have at X – 1, that’s an odd multiplicity because it’sthe first power,
so we are going to cross at X = 1. Sometimes when I’mtalking about those, I like to do something to it to remind me that it’sgonna cross – you might make a little X or something just to tell yourself, whateveryou want to do.
Now for the y-intercept.
end behavior of a function
So this is pretty easy because they-intercept occurs when X is zero, right? So this first factor here turns zero,it’s gonna zero everything else out,
so the y-intercept is at 0 which is alreadyon my graph.
Okay so to graph it, we’re gonna start up, and remember we aretouching at – 2 and touching at 0 and then crossing at 1. So bounce,bounce, cross.
Okay, last one. So, looking at the end behavior, we have anegative and X to the first power, X to the first power, X squared. So that’s gonna give me anegative X to the fourth, which behaves like a negative quadratic, so it’s downon both ends.
We’re not sure what’s happening in the middle, but we’ll bealright. So, looking at our factors for the X + 1 that has an odd multiplicity, right,
of one, which means that we are going tocross at X = -1 and we can go ahead and put a point on ourgraph. For the second factor X + 2, it also has an odd multiplicity of one,so we’re gonna cross at X = -2 and I’m actually going tomake those both X’s so I don’t forget – that’s my brain thing saying that’sa cross.
And then for X – 1 squared, that has an even multiplicity, so that’sa touch or what I call a bounce,
at X = 1, and then we need to look atthe y intercept. So, for the y intercept, again, remember you’re making X equal tozero. So we have a negative,
from the first factor we have a positive 1,second factor a two, third factor a negative one squared.
So we have, let’ssee, a -1 times 2 is -2 times 1 is still -2, soI’ll plot my y intercept at -2 and then we’re ready to graph. So we’regonna start down,
end behavior of a function
and we’re going to bounce, bounce, come down and get thisy-intercept, and – I did it wrong, so I bounced where I should have crossedright? end behavior of rational functions.
And I want to point out that I knew I was wrong because my end behavior wasn’t correct.
So, we’ll go back again – I’m starting down but my X’s mean cross -shouldn’t I know what my own little things mean to me?I guess not. So we’re starting down,
we’re gonna cross, and cross, get oury-intercept, and bounce.
There we go. Alright, I hope you found this helpful!