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^3+3x^2-1

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 negative 10 X to the exponent five plus nine X squared minus 17. We're given four answer choices. Option A as X goes to infinity, F of X goes to infinity. And as X goes to negative infinity, F of X goes to negative infinity. Option B as X goes to infinity, F of X goes to negative infinity. And as X goes to negative infinity, F of X goes to positive infinity. Option C as X goes to infinity, F of X goes to infinity, as X goes to negative infinity, F of X goes to infinity. And finally, option D as X goes to infinity, F of X goes to negative infinity. And as X goes to negative infinity, F FX goes to negative infinity. Now we have our function F of X which is equal to negative 10 X to the exponent five plus nine X squared minus 17. And the end behavior of this graph we can determine just from the leading term. So our leading term is going to be the term with the highest exponent. OK. And in this case, that's negative 10 X to the exponent five. Now we want to write the degree of this polynomial, the degree of this polynomial is five. OK. That's the highest exponent we have and five is an odd number. And we want to note the sign of our leading coefficient. OK. Our leading coefficient is this number that multiplies our X to the exponent five term. So our leading coefficient in this case is negative 10. And so it is negative. So we have a polynomial, it is an odd degree polynomial with a negative leading coefficient. What that means is I'm gonna draw a little graph. So we have our X and Y axis just a little sketch. What that means is that in the second quadrant, we're gonna have the function going up to the left and in the fourth quadrant, it's going down to the right. OK. And this helps us think about the end behavior. And we can see from our diagram, OK? And thinking about this, if the function is going up to the left out of quadrant three or sorry out of quadrant two, that means as X is going to negative infinity F of X is going to positive infinity. Now, similarly, it's going down to the right and quadrant or so as X is going to positive infinity, F of X is going to negative infinity. And this is true for any odd polynomial. OK, odd degree polynomial that has a negative leading coefficient. Now, it can be easy to mix this up. Sometimes. What I like to do is think about functions, you know, OK, think about X squared. If you're looking at an even polynomial, think about X cubed. If you're looking at an odd degree polynomial, which way does that go? And that rule is gonna hold steady for all of these odd degree polynomials with negative leading coefficients. All right. So compare to answer choice as X goes to negative infinity, F of X goes to positive infinity. And as X goes to positive infinity, F of X goes to negative infinity, this corresponds with answer choice. B 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)=-4x^3+3x^2-1

Key Concepts

End Behavior of Polynomials

Leading Coefficient

Degree of a Polynomial

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