Let ƒ(x) = (x - 3) (x + 3)²

g. Use your work ...

Question

Let ƒ(x) = (x - 3) (x + 3)²

g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

Accepted Answer

Welcome back, everyone. In this problem, we want to draw the graph of the function F X equals X minus 1 multiplied by X + 1 squared. And here we have a graph on which we're going to put F of X. Now to draw the graph of FF X, we need to determine a few key properties that will help us to plot. Let's try to go through each. First, let's start with the domain of FFX. Now notice here that FF X is a polynomial and the domain of a polynomial function is all real numbers. So in interval notation, the domain is going to be open parentheses negative infinity, infinity closed parentheses. Next, we need to check our our function for symmetry, OK? And to check for symmetry, we need to see if the function is even, odd, or neither. Now, how can we tell each of those? Well, a function is even, OK, function is even if F of negative X is equal to F of X for all values of X, OK? Now in this case here, that means if we try to find F of negative X, it's going to be equal to negative X minus 1 multiplied by negative X + 1 squared, OK. And no, it's gonna be equal to negative X + 1, OK, multiplied by X minus 1 squared. Which is not equal to F of X, OK? Therefore, our function is not even. And now if we think about it, FF neither is F of X of negative X equal to negative FFX. And that's what we can tell if a function is a. A function is odd if F of negative X equals negative F of X for all values of X, but again, we can tell that that is not true. So a function is neither even nor odd. OK, let me write that in red here. It's neither even nor odd. Now what else do we need? Well, we'll need the function's critical points. And to do that, we can find the first derivative of F of X, OK? Now, we need it because at the critical points, OK, the first derivative. is going to be equal to 0. So, is that the case? Well, let's differentiate FF X, OK. Now, notice it's a product, so we can differentiate it using the product rule, OK. And when we do that, we should get the first derivative to be equal to. 1 multiplied by X + 1 squared + 2 multiplied by X + 1 multiplied by X minus 1. And if we simplify that, OK, then we should get that it equals X + 1 multiplied by 3 X minus 1. This is the first derivative. Since at a critical points, the first derivative equals 0, let me come to the left here. That means we can say that at our critical points, X + 1 multiplied by 3x minus 1 is going to be equal to 0, which implies that X is either going to be, well, X is going to be -1, and X can also be 13. So there are 2 critical points at X equals -1 and 1/3. Now that we have our critical points, to draw our graph, we need to figure out on which interval or function is increasing or decreasing. So what are our intervals? Well here, OK, if we think about our X-axis of our graph. OK. Then if our critical points are at -1 and 1/3, then we have infinity and negative infinity, here we have 3 intervals that we're concerned with, where X is between negative infinity to -1, from -1 to 1/3, and then from 1/3 to infinity. So let's analyze each of those intervals, and we can use a sign chart to test the values here, OK? No. To determine, so let's write that out to determine. Where FFX is increasing or decreasing. OK. We know that our function will be increasing if the first derivative is positive and decreasing if the first derivative is negative. So for our interval. Between negative infinity to -1, we can pick a value for X and substitute it into the derivative of FF X to figure out if it's positive or negative to tell its sign. So let's use when X equals -2, OK? Then our first derivative. Is gonna be equal to -2 + 1 multiplied by 3 multiplied by -2 minus 1. That equals 77 is greater than 0, so at this interval, our function is increasing. No, we're just gonna repeat that for the other two intervals. Or repeat that process. Now, for X in the interval between -1 and 1/3, OK, what value could we use? Well, let's say that X equals 3, X equals 0, sorry. When X equals 0, then our derivative is going to be equal to, of course, 0 + 1 multiplied by 3 multiplied by 0 minus 1. Which is gonna be equal to -1, OK. And the -1 is less than 0. So at this point we know our function is, or in this interval, our function is decreasing. Finally, For the interval from 1/3 to infinity. Let's set X to be 1. OK. Then our first derivative is going to be 1 + 1, multiplied by 3 multiplied by 1 minus 1, OK. And that's gonna be equal to 4. It's increasing on this interval. So now we know what's happening at each of our intervals based on our critical points. No, where do we go from here? Well, next we can determine the concavity of our uh graph, OK? In order to determine the concavity, we can use our uh our 2nd derivative, function's second derivative, OK? Now, here. Let me put this in black, OK. The second derivative. Of F of X, we differentiate our function again, OK? We should find that it's gonna be equal to 1 multiplied by 3X minus 1. OK. Again, it's a product, so we can apply the product rule of differentiation. Plus 3 multiplied by X + 1. And when we simplify it here, we should get this to be equal to 6 X plus 2. And no To find those critical, well, here for our second derivative, to find, rather, um, where it is concave down or up, we can analyze the sign of the second derivative, OK? Now, for X. OK, for X less than 1/3, we can pick a value for X, OK. Let's use an X E equals -1. And when X equals -1, our second derivative. Is gonna be equal to 6 multiplied by -1 + 2, which is -4. That means it's negative, it's less than 0. So FFX is concave done. Yeah, I should have probably written that in red here. At this point we know FFX is going to be. Concave don't. On the other hand, OK, for X being greater than -1/3, OK. Because now, we get that X value if we solve for a 6 X + 2 equals 0, so that means 6 X would be negative2, and thus X would be -2 divided by a 6 or 1/3. I think I'll just make a note of that. Uh, just, just in case I didn't say that earlier, my apologies. X would be -13. That's why we're using this value -1/3 here. And know when X is greater than -13, then we could use a value on that interval when X equals 0 or second derivative of 0 is going to be equal to 6 multiplied by 0 + 2, which equals 2, that's greater than 0. So F of X is going to be concave up. So now we know apart from when it's increasing and decreasing, when our function is going to be concaved on our oh, now we know our functions concavity. Now, if we think about it here. We also, we found our critical points, but we need to know which ones are the maximum and minimum. And we can figure that out using our information of whether FFX is increasing or decreasing. Notice here that F of X is increasing before -1, but decreasing after -1. So XT equals -1 is our maximum, OK? Let's put that right here. On the other hand, F of X is decreasing before 1/3, but increasing after 1/3. So it means that X equals 1/3 is our local minimum point, OK? And finally, For our inflection point, notice that at X equals -1/3 or inflections, or, or concavity changes. So X equals -1/3 is our inflection point, OK? That's our inflection point. Now, what else do we need to plot our graph? Well, we'll need to know our graphs and behavior, OK? How can we figure that out? Well, We know that as x approaches infinity F of X will also approach infinity and as x. Approaches. Negative infinity. OK. FFX also approaches infinity because the leading term X cubed dominates. Finally, for our intercepts, uh, we know that our X intercepts occur with FFX equals 0 and in the interest of time. Uh, or those X intercepts will be equal to -10 and 10. OK. We can find those by solving in F of X equals 0, but on the other hand, for our Y intercepts, we know it will occur where X equals 0. So a Y intercept is 0-1. So let's do a quick check of what we know. We know our Y intercepts, or X intercepts, when our graph is concave up and concave down, the intervals on which it is increasing and decreasing. We know the inflection point. Uh, we also know our maximum and minimum points. We know our graph is neither even nor odd, and we know the domain of our graph. So with all this information, let's plot it. So as we said, We know that -10 will be our local maximum. OK. Uh, we know that negative, 0-1 is our Y intercept and 13, which is probably gonna be somewhere about here, 1/3 -3237, OK, is gonna be our local minimum. 3227, 3227, sorry, and 10 is our other X intercept, the next X intercept. So with this information, now we can plot our graph and it's gonna look something like this, OK? Let me draw that curve a little bit better, just, uh, just, just. Just to make it look a little more accurate, OK, but when you draw your curve, it should look something like this. Just give me one more try, OK? I think it's because my uh. My, my, my having a little issue with my. And so here, OK. But this is the last one, OK guys, this is it. This is it. Whatever we get, this is it. So our curve should look something like that. This is gonna be the graph of FFX. Thanks a lot for watching everyone. I hope this video helped.

Let ƒ(x) = (x - 3) (x + 3)²

g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

Key Concepts

Factoring Polynomials

Multiplicity of Roots

End Behavior of Polynomials

Watch next