 Another type of graph that it'll be important to look at are polynomial graphs, or, which is to say, the graph of a polynomial function. And so to understand how to graph a polynomial function, well, then we might take a look at the following. When we graphed linear functions, we started by finding the x and y intercepts. When we graphed quadratic functions, we started by finding the x and y intercepts. So when we graphed polynomial functions, we'll want to do something completely and totally different. We'll start by finding the x and y intercepts. Now, where things change are that for a linear function, the x and y intercepts are enough because this is the graph of a straight line, and I have two points. And once I have two points on the line, I can graph the line. For a quadratic function, things are a little bit more complicated. And for a polynomial function, we have to do even more things after we find the x and y intercepts. But what we want to do is we want to determine the behavior between the intercepts. We'll talk about what that means. We want to determine what's called the end behavior of the polynomial function. And to obtain our graph, we'll do a connect the dots graph that will incorporate all of this work that we've done. We've found the x and y intercepts. We should use it. We've determined the behavior between the intercepts. We should use it. We'll determine the end behavior. We should use that as well. And for aesthetic purposes, this connect the dots graph will have a whole bunch of corners in it. We'd like to round those corners off. How we do that more precisely is something you'll study in calculus. So for now, let's take a look at a graph like y equals this thing. y equals x plus three squared times x plus two to the fifth. And we want to find the x and y intercepts. The x intercepts are going to be the places on the graph where y is equal to zero. So I'll use the equation for the graph. I'll set y equal to zero and solve for x. So here I have zero is right hand side is a product of two things. So I have product equal to zero. So I know that one of the two things is going to be zero. So I know that either x is equal to minus three or x is equal to negative two. And this gives me the x coordinates of the x intercepts. And so I know where they're located. For the y intercepts, I want to find the places on the graph where x is equal to zero. So again, I'll use the equation of the graph. I'll allow x to be zero and solve for y. And I'll evaluate this. That's three to the second, nine, two to the fifth is 32. Product nine by 32, 288. And this gives me the y coordinate of the y intercept. And so I know where the y intercept is located. Now with a polynomial graph, we want to do a little bit more here. So the next thing we might want to do is we want to figure out the intermediate behavior. And that involves figure out what the graph looks like between the x intercepts. So we've already determined where the x intercepts are. And the important question here is whether the graph is above or below the x axis between the intercepts. So we could determine this by looking at the sign of the y values. Remember our graph y equals f of x. Our y values tell us the vertical displacement, y is positive, we're above the axis, if y is negative, we're below the axis. So I know where the intercepts are because I figured those out already. And I want to know what's going on between these two x intercepts. We don't actually need the y intercept for this. What's important is this x intercept here. Now I could pick a test point someplace in between the two x intercepts. But really I only care about the sign of the y value. If the y value is positive, I know I'm going to be above the x axis. If the y value is negative, I know I'll be below the x axis between the intercepts. So rather than worrying about a magnitude, rather than calculating an actual y value, let's focus on the signs. So here's how we can go about this. Here's why it's nice to have these graphed out. If I'm in between negative 3 and negative 2, my first factor here is x plus 3. So I'm at a point here and I add 3. And if I add 3 to my x value, that's going to put me over here someplace. So x plus 3 is going to be a positive number. I'm going to square it. Well, that'll still make it positive. x plus 2, on the other hand, if I'm at a point in this interval, if I add 2, because this point is at negative 2, if I add 2, I don't quite make it to the origin. So that means that x plus 2 will be negative. And because I'm raising it to an odd power, x plus 2 raised to power 5 will also be negative. And that means this product here is going to be the product of a positive number with a negative number. So this product is going to be negative. And this product is equal to, is the same as my y values. So I know my y values will also be negative in this interval. And that means that in this interval, my graph is going to be down here below the x-axis. And I'll make that observation explicit. And to remind us of the fact, I'll go ahead and drop a point there someplace below the x-axis. So I know that I've got to hit this point down here someplace. What about n-behavior? Another important feature of functions is what's called the n-behavior. What happens to the function values as x gets very large, either a very large positive number or a very large negative number? So for the n-behavior of a function, the important question is we want to know the sign and magnitude of the function values when the input values become very large and positive. We say as x goes to plus infinity or as the input values are negative and large. And again, we say as x goes to minus infinity. So here the plus and minus are based on the sign, positive or negative. The infinity symbol is this concept that the values are getting arbitrarily large. So let's say I want to find the n-behavior of our function. And I want to know what happens to our output values, our y-values, as x gets positive and large. So let's take a look at the analysis here. Again, the signs are more important. The magnitudes are kind of relevant, but the signs are really the first thing we want to start with. If x is positive and large, x plus 3 will be a large positive number plus a small positive number. Well, that's just going to be a large positive number. x plus 3 squared will be the square of a large positive number. That's going to be a large positive number as well. x plus 2, that's going to be a large positive number plus a number. Again, large positive. x plus 2 to the fifth is going to be the fifth power of a large positive number, and that's going to be a large positive number. And so that says that this product will be the product of two large positive numbers. When I multiply two large positive numbers together, I get a large positive number. So if x is positive and large, this product will also become positive and large. We also want to look at the other end, what happens if x is negative and large. So x plus 3 is a negative large number, plus 3 is a large negative number. But when I square it, it will become a large positive number. x plus 2 will be a large negative number, and if I raise it to the fifth power, well that's a large negative number raised to the fifth power, which is going to be a large negative number. And so our product, x plus 3 squared times x plus 2 to the fifth, is going to be the product of a large positive number with a large negative number, and that's going to be a large negative number. So as x goes to negative infinity, this quantity is going to become negative and large. And now we could put everything together. So we want to sketch a graph showing our correct end behavior, x and y intercepts, and behavior between the intercepts. So we've already found the x and y intercepts, so we can plot those points. We already know that the graph is below the x axis on this interval between the two x intercepts. We also know that as x gets large and positive, y becomes large and positive. So we want to graph that. So let's see, we have points that are large positive x, that's way over here to the right, large positive y, that's way up here. So I might graph a point up there someplace. And as x becomes large and negative, y becomes large and negative. So we'll show this. Again, x is large negative way over here to the left, y is large negative way down here below the x axis. So I'll graph that point. I'll draw my stick figure sketch. I'll connect the dots. And again, it's aesthetically pleasing to round this off to give us a smoother graph. So I'm going to round the corners off a little bit, and my graph is going to end up looking something like that.