In Exercises 19–24, use the Leading Coefficient Test to determ...

Question

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f (x) = - 11 x^{4} - 6 x^{2} + x + 3$

Accepted Answer

Hello, today we are going to be determining the end behaviors of the given function using the leading coefficient test. Before we determine the end behaviors, let's quickly review what the leading coefficient test states. The leading coefficient test allows us to determine the end behaviors of a given function based off the leading coefficient and the highest exponent. The test states that if we have an even leading exponent such as 2468 or any other even number. After that, there are two possibilities for the end behaviors. If we have an even leading exponent and we have a positive leading coefficient, then the end behaviors of the graph are going to be increasing on the left and right hand side. In addition to this, if we have an even leading exponent and a negative leading coefficient, then the end behaviors of the graph will be decreasing to the left and right hand side. Also, let's suppose that we have an odd leading exponent if we have an odd leading exponent such as 3157 and any other odd number after that, and we have a positive leading exponent, then the end behaviors of the graph are going to be increasing on the right hand side and decreasing on the left hand side. Furthermore, if we have an odd leading exponent and a negative leading coefficient, then the end behaviors of the graph will be decreasing on the right hand side and increasing on the left hand side. So the leading coefficient test gives us four possibilities for the end behaviors of the function. Now let's go ahead and take a look at the given function. So we are told that F of X is equal to negative 29 X to the fourth minus three X squared plus five X plus six. We want to pay attention to the leading term with the highest exponent. That term is going to be negative 29 X to the fourth power. First, let's take a look at the exponent, we have a leading exponent of four, which means that the, the leading exponent is even. So that tells us that there are two possibilities for the end behaviors. They will either be increasing in the same direction or decreasing in the same direction. We will take a look at the leading coefficient, the leading coefficient is negative 29. So we have an even leading exponent with a negative leading coefficient. That means that by the leading coefficient test, the end behaviors of the graph will be decreasing to both the left and right hand side. And with that being said, the answer to this problem is going to be D the graph falls to the left and right hand side. So I hope this video helps you in understanding how to use the leading coefficient test. And I'll go ahead and see you all in the next video.

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In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f (x) = - 11 x^{4} - 6 x^{2} + x + 3$

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In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=−11x4−6x2+x+3f(x)=−11x^4−6x^2+x+3

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f (x) = - 11 x^{4} - 6 x^{2} + x + 3$