Graphing functions Use the guidelines of this section to make ...

Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 6x² + 9x

Accepted Answer

Welcome back, everyone. In this problem, we want to draw the graph of FF X equals X cubed minus 8X2 + 16 X. For our answer choices, we have different graphs that show our possible answers. So that was A, here is B, here's answer choice C, and then answer choice D. Now, since we already have the graphs, all we need to do is to figure out the critical points or the important points, whether it's the critical point or the points of inflection, uh, or or domain, etc. and then we can do a sketch and compare to our graphs and our answer choices. Now there are a few properties that we'll need to know to help us draw the graph or to identify which is the correct graph. First, let's start by identifying the domain. Now notice here that this is a polynomial and the domain of a polynomial function is all real numbers. So that means our domain is open parentheses, negative infinity, infinity closed parentheses. Next, let's check our graph for symmetry. We want to know if it's symmetric about the X or Y axis, OK? And first, to test symmetry, we can check if our function is even or odd. Our function is even, OK, our function is even if. F of negative X is equal to F of X for all values of X. In other words, if we substitute negative X into F of X and we get back F of X, then our function is even. So we can test that. By saying no that F of negative X is going to be negative X cubed minus 8 multiplied by negative X2 plus 16 multiplied by negative X. Negative XR cubed is negative X cubed, negative X2red is gonna be X2, so this is gonna be minus 8 X2, and 16 multiplied by negative X is -16X. This is not equal to F of X, so our function is not. Even, OK. Our function is not even. Next, we know that our function is odd. If F of negative X, OK, is equal to negative, the negative value of F of X. Now, what is the negative value of FF X going to be? Well, negative F of X is going to be equal to negative X plus 8 X2 minus 16 X. This is not equal to F of negative X, so this also tells us that our function is not odd, so it's neither even nor odd. Now what else can we find? Well, the next thing we can think about is to try and find the critical points of our function, OK. And the critical points of our function are at the point where our first derivative is equal to 0. Now, what's the first derivative of our function? Well, if we differentiate our function with respect to X, the first derivative is 3 X2 minus 16 X plus 16. OK? So we need to figure out the values that are going to make 3 X2 or that is going to make 3 X squared. Minus 16 X plus 16 or first derivative equal to 0. Now what will those values be? Where we can find them using the quadratic formula. Here we know that A will be 3, B is -16, and C is 16. So that means X. By the quadratic formula is going to be equal to -16 plus R minus the square root of -16 squared. -4 multiplied by 3 multiplied by 16. All divided by 2 multiplied by 3. And now if we go ahead and solve for X here, I'll allow you to work out the, the, the finer print on your own in the interest of time, but you should find that you get X equal to 16 plus or minus 8 divided by 6, which means that X is either going to be equal to 16 + 8 divided by 6, which is 4. And X is going to be equal to 16 minus 8 divided by 6, which is 4/3. Thus our critical points are where X equals 4 and X equals 4/3. Now, our first derivative also helps us to determine where our function is increasing or decreasing, OK? And since we know that our critical points are at X equals 4 and X equals 4/3, then we can test the three intervals that are formed here, namely, For so here. Namely, we're testing where X is in the interval negative infinity for 3, OK. Where X is in the interval. 4/34. And where X, OK, let me make some space over here, where X is in the interval 4 to infinity. Now for X being an interval negative infinity to 4/3, we can pick a value there. Let's say we use an X equals 0. When X equals 0, the first derivative, OK, is going to be equal to 3 multiplied by 0 square minus 16 multiplied by 0 + 16, which would be equal to 16. Notice here. That the first derivative is positive, so that means in this interval, our function FFX is increasing, OK. Let's let me highlight that in red as well. No, for the interval where X is between 4/3 and 4, let's use when X equals 3. And now, at X equals 3, OK. The first derivative is going to be equal to 3 multiplied by 3 squared. That's 27, minus 16 multiplied by 3, that's 48 plus 16, OK. And when you work at all, 27 minus 48 plus 16. We get -5, OK. And -5 is less than 0. It's obviously it's negative. So it means that at this point, FF X is decreasing, OK? So F of X is decreasing for our interval, where X is between 4/3 and 4. Now finally, let's test the interval where X is between 4 and infinity. Uh, let's use the value where X equals 5. Now at X equals 5, the first derivative, OK, is gonna be equal to 3 multiplied by 5 squared minus 16 multiplied by 5 + 16. Now you could work the value out, but the point is this value is going to be positive, OK? And when because this value is positive, what does that tell us? Tells us that at this point our function, or on this interval rather, our function is increasing, OK? So we know the behavior of FFX for the different intervals between the critical points. Now, the next thing we can do is to figure out the second derivative. And the second derivative is going to tell us where our points of inflection are, OK? Now, To find the second derivative, we already have the 1st derivative. OK, so we know that for our inflection points. This is where the second derivative is equal to 0, and here, our first derivative is 3X minus 16 X plus 16. So that means the second derivative is going to be equal to 6 X minus 16, OK? And when 6 X. Minus 16 equals 0, then X equals 16 divided by 6 or 8/3. So, when X equals 8/3, that's a point of inflection, OK? No. What do we know about this point? What else does this point tell us? Well, it can help us to determine the concavity of our graph, OK. Now, at X, or sorry, let's say we're gonna do it similar to what we did for increasing points. For X, less than 8/3, OK. We could choose a point, let's say at X equals 0, OK, then our second derivative. Is going to be equal to -16. It's less than 0, so that means at for X is less than 8/3, our graph is going to be concave down. OK. Again, let me highlight that in red. And now, if we pick a point where X is greater than 8/3, OK, let's say we use where X equals 4. Then the second derivative is going to be equal to 24 minus 16, which is 8, and that's greater than 0. So that means for this interval, our graph is concave up, OK? So at this point or at this interval, our graphics can't give up. Now is there anything else we need to know? Well, We need to first locate any other asymptotes and determine the end behavior. And no, since the polynomial is a, since the function is a polynomial, sorry, there are no asymptotes and for the end behavior as X approaches infinity. F of X also approaches infinity and as X approaches negative infinity, then F of X. Also approaches infinity, sorry, negative infinity. Well, let, let me, let me fix that rather here. FFX also approaches negative infinity, OK. And as X approaches negative infinity. That happens because the leading term excubed dominates. And now finally we just need to find our intercepts to help us plot, OK? So now for our intercepts. OK. Already we know that our Y intercept is gonna be equal to 0 because that's where, where the X value is 0 and F of 0 is 0. But for our X intercepts, OK, again, in the interest of time, when you solve, we're gonna set F of X to 0, and when F of X equals 0, then you should get X to be equal to 0 and 4. So now let's do a quick, let's do a quick recap. Our X intercepts are at 0 and 4. Our Y intercept is 0. We know the end behavior of our graph. Uh, we know when it's concave up and concave down, and we are also able to find the local maximum, minimum, and inflection points. OK? So now And as we said, we also know where critical points are that our graph is not even or odd, and we know it's domain, OK? So now, with all of that in mind, then if we were to plot our graph, OK, and again, I'll just do a quick sketch here and then we'll compare them to our answer choices. OK. Then for our graph, if we put the information that we have, we know that it's X intercepted 00. The local maximum is at 4/3 and 256 divided by 27. That would be the Y coordinate when you plug it into our graph. The local minimum is for 0 and that also acts as the X intercept. And using what we know about our graph, we know then when it's increasing, decreasing, and when it's concave up, concave down. So instead, our graph is gonna look something like that, OK? This is gonna be the graph of FFX. I know if we compare that along with the points that we know to our answer choices, then we should be able to tell that answer choice B. Is our correct answer. Y intercept is at 00, local maximum, local minimum. Thanks for watching everyone. I hope this video helped.

Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 6x² + 9x

Key Concepts

Polynomial Functions

Critical Points and Extrema

End Behavior

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