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) = 4x - x^3

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the given function. Were asked to do it through three steps. So the first is to determine the graphs and behavior using the leading coefficient test. The second is to determine if it has y axis symmetry, origin symmetry or neither. And then the third is to actually graph it. Okay, now the function we're given is f of X is equal to negative two X times two X squared minus three. Okay, so when we're looking at the leading coefficient tests, we're looking at part one, we need to consider the leading coefficient and we need to consider the degree of the polynomial. So if we consider the degree of the polynomial, let's just expand this. So we can see clearly we get negative for X cubed plus six X. Okay. And so are Polynomial has degree three. That's the highest coefficient. Or sorry, the highest exponent. We have a degree three polynomial which is odd and our leading coefficient is negative four. Yes, we're leading coefficient Is negative less than zero. Alright. So if we have an odd polynomial with a leading coefficient less than zero, then what happens to F of X? As X goes to positive infinity? Well, as X goes to positive infinity Y or let's say f of X in this case is going to go to negative infinity and vice versa. When x goes to negative infinity Y or f of X is going to positive infinity. Okay, so this is the end behavior of our graph That we found using the leading coefficient test. Okay, so part one is done Now we can move on to part two. Okay, I'm just gonna right in between here of these solutions, does it have y axis symmetry, origin symmetry or neither? Okay, now in this case our function has all of its terms odd. Okay, all terms of f of X are odd. What does this mean? This means that the exponents on every single term is odd. Okay, we have an exponent of three on the first term. We have an exponent of one on the second term. We just have those two terms. Both of those are odd. So what this tells us is that we are going to have origin symmetry. Okay, Alright, so now we know what the end behavior is, like we know what our symmetry is. Like let's go ahead and graph this function. Let's move down here. We're gonna use this space. One thing that will help us with our graph is going to be the X intercepts are the zeros of this graph because we've we've been given this nice function that's already factored. It's fairly straightforward to find those zeros. So let's go ahead and do that. That's just gonna help us out with our graph. Okay, so we get from the first Term that X is equal to zero. Okay, Or we get two, X squared minus three is equal to zero, which tells us x squared is equal to three halves or X is equal to positive or negative square root of three halves. Okay, so we have three roots. We have negative square root of three halves. Zero and positive square square root of three halves. And it makes sense that we have this negative and positive square root of three halves because we have origin symmetry. So we have that same route on both sides. All right. So let's scroll down see if we can get ourselves some space to draw this function. So we have the access here. Now we have our three routes 00, Squirt 3/2, Negative Square Root of 3/2. We know that our in behavior tells us when X goes off to infinity, Why is going to negative infinity? So we know that our function is going to be going down here and opposite. It's gonna be going up here when x is going to negative infinity. This is going to positive infinity. And that also mimics that symmetry about the origin. Okay, so a function is going to do something like this. Okay. Alright, so if we compare that to the possible answers, we see right here that we have the same graph, D R N behavior. The function F is going to infinity as X is going to negative infinity. The function is going to negative infinity as X is going to infinity. We have origin symmetry and our graph looks like this. All right. That's it for this one. Thanks everyone for watching. See you in the next video.

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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) = 4x - x^3