In Exercises 10–13, use the Leading Coefficient Test to determine...

Question

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).]  f(x) = -x^3 + x^2 + 2x

Accepted Answer

Hey everyone in this problem. We are asked to take the polynomial function given. Use the leading coefficient test to determine the end behavior of the graph and to sketch the graph. Okay, now the function we're given is f of X is equal to negative seven X cubed plus two X squared minus four X. Okay. Alright. So let's look at the leading coefficient test. Now the leading coefficient test wants us to look at the degree of the polynomial and the sign of the leading coefficient. Okay, so the degree of this polynomial? Well, that's going to be the highest exponent, which is three. Okay, so we have degree three which is odd. Okay. And our leading coefficient Is the coefficient corresponding to that highest exponent. So in this case our leading coefficient is negative seven which is negative. Okay, less than zero. So our leading coefficient is negative. We have an odd degree polynomial. So the leading coefficient test tells us that our end behavior will be as follows. As x goes off to positive infinity, Why will go off to negative infinity? And vice versa? As x goes off to negative infinity. Why goes off to positive infinity? Okay, we missed the as. Alright, so we have our own behavior. Okay, so on the right side of our graph, our function is going to negative infinity. On the left side of our graph. Our function is going to positive infinity. Now we want to graph this function, we know the end behavior. We know the degree of the polynomial. Let's just go ahead and try to find out a little bit more information before we graph this. Okay, so let's think about intercepts. Those are always very helpful when we're trying to draw a graph. So let's think about the intercepts and we'll start with Y. Now the Y intercept is given one F. Or sorry, when X is equal to zero. So we get F of zero, which in this case is just going to be zero. Okay, we have X in every single term. So we get zero. Alright, what about our X intercepts? Okay, this polynomial hasn't been factored. So we want polynomial F of X to be zero. This polynomial hasn't been factored. But let's go ahead and try now. We can find that we find that we can take an X. We can factor out an X. Because there's an X on every term. So we get F of X is equal to X times negative. Seven, X squared plus two, X minus four, correct. All right. Now, if we look at this and we set it equal to zero in order to solve for these routes. Well, the first term could be zero which gives us X equals zero. Or the second term could be zero. Now, in this case you can use the quadratic formula if you'd like, we are going to have no real roots. Okay, if you try to use the quadratic formula, you're gonna get the square root of a negative. So you have no real roots. Okay, so this means that our functions are polynomial only has one root. So now we know the route. We know the end behavior of the function. Let's go ahead and draw it. Okay, so we have our axes here. Y X. We know we have a route at 00. And we know that's the only route we know as X gets more and more negative, Y gets more and more positive and vice versa. As X gets more and more positive, Y gets more and more negative. End behavior on the left Is going up out of quadrant two and on the right we're going down out of quadrant four. And so our function is going to look something like this. Okay, if we look at the possibilities, we see that this corresponds with answer C. Okay, so we have the right end behavior as X gets positive infinity F of X goes to negative infinity. On the left end. As X goes to negative infinity F of X goes to positive infinity. And the graph looks like the one we've drawn with a single root. Thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with College Algebra?

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x) = -x^3 + x^2 + 2x

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