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)=10x^6-x^5+2x-2

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 11 X to the exponent eight minus four, X to the exponent six plus seven, X squared minus 13. We're given four answer choices A through D. OK. And each of them has a different combination of M behavior as X goes to infinity and X goes to negative infinity. Now, if we look at our function F of X, when we want to know the end behavior of a graph, what we're interested in is the leading term. OK. And the leading term is gonna be the term associated with the highest exponent. Now, our highest exponent is eight. So the leading term is 11 X to the exponent eight. Now the degree of this leading term, the degree of a polynomial is the exponent. They are the highest exponent. So our highest exponent here is eight. OK? We have X to the exponent eight. So this is the degree eight polynomial. OK. Which means that it is an even degree polynomial. The last thing we want to look at is the leading coefficient. Now, the leading coefficient is just the coefficient of this leading term, the coefficient corresponding to the highest exponent. OK. In this case, that's a love uh which is positive, it is greater than zero. And these two pieces of information are what's key. OK? We have an even degree polynomial and we have a positive leading coefficient. Now, if we draw a little sketch of a Cartesian plane, if we have an even degree polynomial that's positive and it means it's gonna be opening upwards. And so in quadrant one, we're gonna have the function going off up into the right. And in quadrant two, it's going off up into the left. What this means is that as X goes to positive infinity, our function F FX is going off to positive infinity. OK? It's growing, it's getting bigger and bigger and bigger and FX goes to negative infinity. The function is actually doing the exact same thing. OK? It's growing and getting bigger and bigger and bigger as we go to the left. Now, if you ever forget or you get confused, I like to think of those basic polynomials that we know something like X squared. OK? We know what X squared looks like. OK? When it's a positive leading coefficient, we know it opens upwards. OK? When it, it, it's an even function, we know what that looks like to find the end behavior and every even polynomial even degree polynomial that has a positive leading coefficient is going to follow that same pattern. And so the end behavior here as X goes to infinity or as X goes to negative infinity, the function F of X is going off to positive infinity which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=10x^6-x^5+2x-2

Key Concepts

End Behavior of Polynomials

Degree of a Polynomial

Leading Coefficient Test

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