Hey, everyone. You now know absolutely everything that you need in order to fully graph a polynomial function. So here, we're going to put all of that together in order to graph a polynomial function from scratch. Now it might feel like a lot and that's okay. I'm going to walk you through it step by step. And if you follow these steps, you'll be able to graph any polynomial function correctly every single time. So let's not waste any time here and get straight into graphing.
So the polynomial function that I have here is x3 - 6x2 + 6x - 2. Now let's start at step 1 and find the end behavior of our polynomial function by looking at our leading coefficient anxn, which in this case is 2x3. So first looking at our leading coefficient an, this is a positive 2. And because this is positive, that tells me that my graph is going to rise on that right side which I can go ahead and sketch here. Then looking at the degree of my polynomial it is 3 which is an odd number which tells me that the behavior on the end is going to be the opposite. So sketching that on my graph if my right side was rising, the opposite on the left it is going to fall.
Now let's go ahead and move on to step number 2 and find our x intercepts and their behavior. So here we want to go ahead and solve f of x is equal to 0. Now you might have to factor this on your own sometimes, but we already have this pre-factored here so let's go ahead and solve for x. So I'm going to take this factor, x - 1, and simply set it equal to 0. Now if I add 1 to both sides, it will cancel, leaving me with x is equal to 1, and that is my single x intercept, x = 1. Now what is the multiplicity of this x intercept? Well, remember, it's just the number of times that our factor occurs. So here that is 3, which is an odd number. So that tells me that it is going to cross the x axis at that point. So plotting that on my graph at x = 1 I know that it's going to fully cross the axis at that point.
Let's move on to step number 3 and find our y intercept. So our y intercept, we're going to go ahead and compute f of 0 by plugging 0 into our original function right here. This leaves me with 2 times -1 cubed, which is really just 2 times -1, which is simply -2. So this is my y intercept that I can go ahead and plot on my graph. So my y intercept is at -2.
Okay. So we've already plotted a bunch of stuff. We have a bunch of known elements. Let's go ahead and move on to step number 4, which is going to be to determine our intervals where we're not quite sure what's happening yet and then plot a point in each of them. So let's determine our intervals by going through our graph from left to right. So looking at that left side and going until I reach my first known point, my first known point is actually my y intercept. So from negative infinity to 0 is going to represent my first interval of unknown behavior. Then going to my next known point here from 0 to 1, which it's really close together, but that's my next interval, 0 to 1. And then, finally, going from 1 and beyond because I don't have any other known points there, from 1 to infinity is that final interval here.
So now let's find a point in each of these intervals in order to get a better picture of what's going on in our graph. So in this interval, negative infinity to 0, remember I want to choose something that's on my graph. So here I'm going to go ahead and choose x is equal to -1 to get that point. Then from 0 to 1, this is a rather small interval, so it is okay to choose a fraction here. I'm gonna go ahead and choose \(\frac{1}{2}\) in that interval. Then from 1 to infinity, remember, again, I don't want to choose anything too crazy. That's not actually going to help me. So I'm simply going to choose the point x equals 3. Now you can go ahead and so I'm simply going to choose the point x equals 3. Now you can go ahead and pause here and plug each of these into your function in order to get f of x and then come back and check that you've got the same answer as me. So here, for x equals -1, if I plug -1 in, I'm going to end up with -16. So the first point that I can plot here is -1, -16. Then \(\frac{1}{2}\), if I plug that into my function, I will end up getting a -\(\frac{1}{4}\), which I can also go ahead and plot on my graph. Now I know that that doesn't look like it helps a ton here, but it was an unknown interval. And then lastly, I have this x = 3, which if I plug in, I end up getting positive 16, which is the last point that I'm going to plot here, at 3, positive 16.
Okay, now we have a ton of information about our graph and it looks like we can go ahead and move on to step number 6 and simply connect all of our points with a smooth and continuous curve because it's a polynomial function. So starting with the point that I have up here I'm gonna go ahead and connect with a smooth curve all of my points. Now you'll notice that I didn't go through those original end behavior lines and that's totally okay. I still am matching the end behavior. I just found out some more information so I can sketch it more accurately.
So we're completely done. We have finally fully graphed our polynomial function. Let's perform one final check here using our turning points. So with our turning points, remember, the maximum number we can have is our degree minus 1. Here my degree is 3, so 3 minus 1 gives me a maximum number of turning points of 2. Now looking at my graph, does this have more than 2 turning points? No. It doesn't even have one turning point so my last check is good and I am completely done. Now you have fully graphed your polynomial function, let's get some more practice.