Use an end behavior diagram, , , , or , to describe the end be...

Question

Use an end behavior diagram,  ,  , , or  , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=4x^7-x^5+x^3-1

Accepted Answer

Hey, everyone in this problem, we're asked to determine the end behavior of the graph of the following function. The function we're given is F of X is equal to eight X to the exponent five minus two, X to the exponent four plus nine X cubed minus 21. We're given four answer choices, options A through D, each answer choice contains a different combination of the end behavior of the function F of X as X goes off to either positive or negative infinity. Now, when we're looking at the end behavior of the graph, what we wanna do is first look at the degree of the polynomial we have now recall that the degree of the polynomial is gonna be the highest exponent. Now, in this case, the highest exponent is five. And so the degree of this polynomial is five, which is an odd number. The other thing we want to look at is the leading coefficient and the leading coefficient is gonna be the coefficient corresponding to the highest degree term. So our highest degree term is X to the exponent five that has a coefficient of eight. OK, which is a positive number. It's greater than zero. And so the important pieces here are that we have an odd degree polynomial with a positive leading coefficient. What that tells us is that as X goes off to positive infinity, our function F FX is going to go off to positive infinity. And as X goes off to negative infinity, our function F of X goes off to negative infinity as well. And a little trick if you get stuck with a problem like this, and you can't remember which way this goes is to think of a function that you know the behavior of. OK. So think about say X just a line, a linear function, that's an odd degree polynomial. You know that if X has a positive leading term, that means it has a positive slope, it's gonna be going up into the right in quadrant one and down into the left in quadrant three. OK? Which gives us the end behavior that we found as X goes to infinity F of X goes to infinity and as X goes to negative infinity, F FX goes to negative infinity. OK? And this holds true for any odd degree polynomial with a positive bleeding coefficient. If we compare this to our answer choices, we see that this corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with College Algebra?

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=4x^7-x^5+x^3-1

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