In Exercises 19–24,

(a) Use the Leading Coefficient Test to dete...

Question

In Exercises 19–24,

(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither.
(c) Graph the function.

f(x) = x^3 - x^2 - 9x + 9

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the given function. Okay. And we're asked to do it through three steps. The first is determine the graphs and behavior using the leading coefficient then to determine if it has y axis symmetry, origin symmetry or neither. And the third is to go ahead and graph it. Okay, so the function we're given is x cubed plus two, X squared minus three X minus four. Okay, so let's start with part one Now. Part one. We're looking at the leading coefficient test. Now when we're looking at the leading coefficient test, we want to consider the degree of the polynomial in the sign of the leading coefficient. Okay, so the degree here, the highest exponent is three. Our highest exponent is three. We have a degree three polynomial. All right, let me highlight that again. This is odd. Okay, so three is an odd number. So this is odd. We also know that the leading coefficient, the coefficient associated with that highest exponent is just one. It's positive. So it's bigger than zero. Okay, so if we have an odd polynomial with a leading coefficient that is positive. What we're going to have is as X goes to positive infinity, why is also going to go and let me write this instead of Why is F of X? Like the function we're given is going to go to infinity. This X goes to negative infinity. We're gonna have f of X. Also going to negative infinity. Okay, so the end behavior on the right is gonna be off to positive infinity. The end behavior on the left is gonna be off to negative infinity. Alright, so we figured out the un behavior based on the leading coefficient test. Now let's talk about symmetry. This function that we're given has a mix of odd and even terms. Okay, what do I mean by that? What I mean is that there are both odd and even exponents? Okay, we have X cube and X terms that have odd exponents. We have an X squared term as an even exponent. So we have this mix of odd and even terms. And so this tells us that we're going to have neither type of symmetry. Okay, we're not gonna have y axis or origin symmetry. No symmetry here. Okay, now, third thing we want to graph this. Well, we know the end behavior. We know the symmetry. Let's just get a little bit more information so we can have kind of a more accurate sketch of this function. Let's first start with the y intercept. Where is this going across the y axis. This is gonna happen when X is zero. So f of zero is going to equal to negative four. So we have this Y intercept there. We want to talk about X intercepts. If we want to talk about the zeroes of this polynomial we can try to factor, let me just write X intercepts here. So let's try to factor. And what you can see off the back is that we have a factor of X plus one. Okay. And if we divide this leaves us with the function X squared plus x minus four. Okay, so we know we have one root at X equals negative one. Okay, you can go ahead and use your quadratic equation or factor this second part and figure out the rest of the X terms. I'm going to show you that from the information we have is going to be enough. Okay, so let's go down here, give ourselves some room to write out or draw out this function. So we have our access here. Y X. Okay, we know that as X is getting really big, why is going to get really big and vice versa? When x gets really negative goes to negative infinity, Y is going to negative infinity. Okay, so we're going to be going off in these directions. What else do we know? We know that there is no symmetry. We know we have a Y intercept of negative four. Okay, so we have a point that crosses here, we have a positive route or sorry, a negative root at X equals negative one. So you'll see now that this is enough because we know we have to go off to infinity on the right hand side like this, to infinity, we know we have to cross this route and then we need to come back in order to go down to negative infinity. Like our own behavior tells us. Okay. And so we end up with this graph that kind of looks something like this or this is F of X. So if we're going back up and looking at our possible answer choices, well, we'll see is that we found and behavior where F of X goes to negative infinity as X goes to negative infinity and F of X goes to infinity as X goes to infinity, we found that there was no symmetry in our graph with something like the graph shown here. Okay, so we have answer A for this one. Thanks everyone for watching. I hope that video helped see you in the next one.

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In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x) = x^3 - x^2 - 9x + 9