 It is going to be our goal to visualize the basic picture of the graphs of polynomials. Now, we're gonna try to graph these things without the use of technology whatsoever. Now our pictures are not gonna be absolutely perfect, but that's actually a good thing because sometimes when you draw things to scale, it can actually hide the behavior because the extremes on the top and bottom of the range can be so large that it can be very difficult to see it perfectly. So we wanna graph polynomials in such a way that we understand the basic shape of the graph and not just the shape, but also why does the shape look the way it does as opposed to some other picture? When graphing polynomials, you should remember that the graph of a polynomial is always gonna be continuous, which remember what this means here, a continuous graph will have no gaps in it, no holes, no rips, no asymptotes, it's continuous. You could draw your picture with one continuous stroke of your pen, assuming you knew what you were trying to draw at the time. So graphs of polynomials are always continuous. Another thing that's true about polynomial graphs is that they are smooth. And I don't mean like Michael Jackson's smooth criminal or anything like that. What I mean is there's no sharp corners or cusp of any kind, things like that. No sharp things. Ouch, if you touch the polynomial, you aren't gonna poke your finger. You're petting a little bunny, you're not petting a porcupine. So as you draw your polynomials, make sure that the turns are rounded, not sharp zigzaggy things. That's not acceptable when graphing a polynomial. So to further understand the graph, we're gonna be employing the ideas of in behavior, turning points and behavior near x intercepts to graph these things. Also we'll use information with the y intercept. So consider the function f of x equals x plus two times x minus one times x minus three. So what we can see here is that if you were to multiply this thing out, some stuff you would see as the following. If we multiplied out the leading term would be x times x times x, which is x cubed. There's gonna be a bunch of other things. But then if we look at the constant term, because we got the leading terms by taking together all of the biggest powers of x, x times x times x. If we take the product of all the constant terms, two times negative one times negative three, we see that the constant term is gonna be a six. And although there's other terms inside of the polynomial, we care about the leading term and we care about the constant term here. Cause what this tells us is the following. The leading term, x cubed here gives us the in behavior. This is x cubed here. So it'll point up on the right-hand side. It'll point down on the left-hand side. So we see by the in behavior as x goes to infinity, y goes to infinity. And as x goes to negative infinity, y goes to negative infinity as well. So that's information that we need here. We also learn from the constant term that the y-intercept is gonna be six. So we're gonna get some point that's above the x-axis here. Now you'll notice that I've drawn no scale on the x-axis or on the y-axis. And that's because my goal in graphing these polynomials is not to come up with a picture that's perfectly drawn to scale. I wanna get the intuitive idea of what this shape is gonna look like. Cause after all, we're unable to, we can't find a perfect picture at this moment because there's some information about the turning points that we won't be able to determine. We'll come back to those in a second. So we know the y-intercept would be six, which is the positive number. We know the in behavior. What can we say about the x-intercepts? The x-intercept by the factorization above is gonna be negative two, one and three. Notice you always grab the opposite sign. Since you see a plus two, negative two is the root. Because you see a negative one right here, positive one is the x-intercept. And the same thing for the negative three right here gives us a positive x-intercept. Each of these things shows up once and so their multiplicities are one. Each of these have odd multiplicity and odd for us means that we're going to cross the x-axis. So what I would do is I'm then gonna plot some points. So negative two would be somewhere over here. We're gonna have a one and we're gonna have a three as our x-intercepts. And we certainly has a good idea to label these things. One, three, negative two, negative six is our coordinates there. And so we know that we're going to be crossing at these x-intercepts. So I'm gonna use this to my advantage here. So I'm exaggerating my in behavior right now. So what I wanna do is basically the following. I like to start at the y-intercept. So because you're at the y-intercept, I know that I have to at some point come down to the x-intercept as I move to the left. That's gotta happen. Now, how do I do that? I could be like really wiggly, right? But polynomials don't wiggle that much, right? The turning numbers are gonna be minimal because our leading turn is a cubic. This is a degree three polynomial. That means at most I have two. You have less than or equal to two turns on this graph here. So as we graph this thing, we're gonna keep the turns minimal. So we're gonna have to have one turn as we go from positive y-intercept down to the x-intercept. And because we have odd multiplicity, we're gonna cross and go to the other side. And so continuing this thing on, we're gonna go and match up with our in behavior that's on the bottom left of the screen there. Likewise, at the y-intercept, I have to also continue down towards the x-intercept at one. And I'm gonna cross to the other side of the x-axis. As I get closer to x equals three, I'm gonna again have to cross the x-axis and come upward towards my in behavior, which remember my in behavior should be pointing up on the right, the upper right-hand side. And so this then gives me the picture of my polynomial. It's gonna have this shape where it goes up and then down and then up again. This is the basic shape, passing through the intercepts x equals negative two, y equals negative six, x equals one and x equals three. And so let's compare this with, I might have to zoom out one more time. Let's compare this with computer generated image. Oh, well, I thought I could get them all on the same screen that's not gonna happen here. So let's look at this one as actually one drawn by a computer comparison. I'd say we did pretty good, right? Now, some things I should mention here is that our picture is not necessarily perfect, right? Because some things we don't know, we don't actually know where the turning points are gonna be. I mean, we can see them on the computer image. It seems like there's some turning point close to two and one close to negative one, but I can't exactly guarantee that at this venture right here. Turns out that we have enough information from the factorization of the polynomial to find the x and y intercepts with their multiplicities and behavior. But we don't have enough information to find the turning points. In order to do that, we have to actually use calculus. We use something called the derivative, which is a topic we are not gonna do in college algebra. Because again, that takes us beyond the scope of this class. Our picture is not meant to be perfect, just intuition of what the graph is gonna be doing. And you can see that even without the graphing calculator that we were able to get a pretty good, accurate picture of the basic shape of this graph. All right, let's do another example. This time, let's consider the function g of x equals x plus, or two x plus one times x minus three squared. So notice here that if we were to multiply this thing out, we would take a two x and we have times that by x squared. So the leading term is gonna look like two times x cubed. You have to put all the powers of x together. So I have a one x, I have a two x here. So you get an x cube when you're done. And then key track of the coefficient two as well. So the leading terms can be two x cubed. If we focus on the constant term, you're gonna get a one times a negative three squared. So that ends up being a positive nine, one times negative three times negative three. This gives us some important information about the function in terms of in behavior. Our behavior is gonna look like an odd monomial function with positive coefficients. So just like we saw a moment ago, x is gonna go, as x goes to infinity, y will go to infinity. So we point up on the top right and as x goes to negative infinity, y will likewise go to negative infinity. So the graph's gonna point down on the bottom left. Our y intercept is gonna be positive this time. So again, we're gonna get something above the x axis there. Now let's look at the roots of the polynomial because we have an x plus one, our two x plus one that tells us the x intercepts, the first one's gonna be negative one half. Because really all we're doing is we're just solving the equation two x plus one equals zero. That means two x equals negative one and x equals negative one half. This is why you switch the sign because when you move it to the other side of the equation, it'll switch at signs. So we get x is negative one half. That's our first x intercept and the other x intercept will be positive three. In terms of multiplicity, we see that two x plus one, it shows up as a factor of once. So negative one half will have a multiplicity of one. x minus three shows up twice so its multiplicity will be two. The first one has an odd multiplicity, which odd means it's gonna cross the x axis. But x equals three, that's an even multiplicity and this tells us that our function is going to touch the x axis but not cross it. In particular, what we see is the following, that as x approaches negative one half, our function will look like, let me scroll this up a little bit, our function will look like f of x will be approximately, we're gonna plug in negative one half into the three spot. That is we're gonna plug in negative one half in this sector right here and leave the other one alone. So we have a two x plus one and we times that by, we're gonna get a negative one half minus three squared. Now be aware here that if I take a negative minus a negative, that's gonna be a negative. I don't actually care what it is. And when then you square it, you're gonna get a positive. And so this thing will look approximately like some positive number a times two x plus one. It's gonna look like a line. So when we come over to the picture of x equals negative one half, it's gonna look like a positive increasing line. So you get something like that. We can see that already. And I'm actually gonna use a different color to illustrate what's going on here. We expect it to go up like that. And that agrees with the in behavior. We kind of expect it to be connected in a picture like that. The next thing to mention is that as x approaches three, we see that our function will look like, it'll be approximately plug in three for the x, except for where it says x minus three, you're gonna get two times three plus one times x minus three squared. Again, that's gonna be a seven, but all I really care about is that it's a positive number. The exact value doesn't matter. You're gonna get a positive seven x minus three squared. What this tells us is that when we come over to x equals two, our function will look like an upward concave parabola, something like this. Now personally, when it comes to the x intercepts and the behavior near the x intercepts, I generally don't plug the values into the function to get these exact coefficients because the information I have already gives me that. When I combine information about touching and crossing, because this is gonna be a cross and this is gonna be a touch, if you combine that with the y intercepts location and the in behavior we already know, that information is redundant. Now don't get me wrong, redundant information is wonderful in this situation, but we don't exactly need it. We have enough information as it is. So what we're gonna do is the following. So we have an x intercept at negative one half. We have an x intercept at two and we have a y intercept here at nine. And so starting at the y intercept, if we go downwards, we're gonna cross a negative one half because that's an odd multiplicity and also because our in behavior wants us to go down over here. So if we draw our picture, we're gonna kind of come down, down, down, down. We get something like that. On the right hand side, we know that as we come towards x equals two, we're gonna have to kind of just bounce off the x-axis because we touch the x-axis, we don't cross it. Then we have to come upward to match the in behavior we have. And so this then gives us a picture of the graph. We have a crossing at x equals negative one half. We have a touching at x equals two. We only have two turns on the graph because this is a cubic function so it can have at most two turns. And two are gonna have to be necessary. On the upper right, it goes up. That is on the right side, it goes up. On the left side, it goes down. This matches the information on turning points, intercepts and in behavior. This is giving us a pretty good picture of this function. Let's scroll down to see a computer-generated image. Here we go. Again, it looks pretty close to what we drew earlier. The exact location of the extrema, we don't exactly know. Now the fact that x equals three, since it touches the x-axis and doesn't cross, that does have to be extrema. It's gonna be a local minimum in this situation. So that one we do know. But the location of the local maximum, we have no idea where it is. It could be, we might guess one, but one might be somewhere right here. It actually looks like it might be a little bit less than one. Also, what about the point of inflection? Where is that? At this venture, we're not worried about the point of inflection. Because again, the inflection points and the turning points, the extrema, the extrema here, we know that they exist. We know roughly where they're gonna be. But if we wanna find the precise location of those points, we're gonna have to use derivatives from calculus. So that's a topic we would have to do in the sequel to this class. At SUU, this is math 12, 10, calculus one. But for this, even without any calculus, just our pre-calculus tools, we have enough information using endpoints, x-intercepts, y-intercepts, and turning points to actually get a very good picture of what our function looks like. And that'll be good enough for our purposes here in pre-calculus.