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. $f (x) = 5 x^{4} + 7 x^{2} - x + 9$

Accepted Answer

Hello, today we are going to be using the leading coefficient test to determine the end behaviors of the given function. Before we look at the given function, let's quickly review what the leading coefficient test states. So leading coefficient test allows us to find the end behaviors of a graph based off of the sign of the leading coefficient. And the leading exponent it states that if we have an even leading exponent such as an exponent as 2468 or any other even exponent, if we have this even exponent and the leading coefficient is positive, then the end behaviors of the graph are going to be increasing to both the left and right hand side in the same direction. In addition to this, if we have an even leading exponent and a negative leading coefficient, then the end behaviors of the graph are going to be decreasing in the same direction. The test also states that if we have an odd leading exponent such as 1357 or any other odd number after that, and we have a positive leading coefficient, then the end behaviors of the graph are going to be increasing to the right and decreasing to the left. In addition to this, if we have an odd leading exponent and a negative leading coefficient, then the end behaviors are going to be increasing to the left and decreasing to the right. So the leading coefficient test gives us four options to choose from based off the leading term in the function. Now let's go ahead and take a look at the given function. The problem tells us that we have the function F of X is equal to two X to the fourth power plus 19 X squared minus three X minus two. We are going to take a look at the leading term with the highest exponent. The leading term with the highest exponent is two X to the fourth power. First, let's take a look at the leading exponent. The leading exponent is four, which means that we have an even leading exponent. This means that the end behaviors are going to be increasing in the same direction or decreasing in the same direction. Now let's take a look at the leading coefficient. The leading coefficient is positive too, since we have a positive leading coefficient with an even exponent, that means that the end behaviors are going to be increasing to both the left and right hand side. Since the end behaviors are going to be increasing in the same direction on both the left and right hand side. The answer to this problem is going to be c the graph rises to the left and right. 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. $f (x) = 5 x^{4} + 7 x^{2} - x + 9$

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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)=5x4+7x2−x+9f(x)=5x^4+7x^2−x+9

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f (x) = 5 x^{4} + 7 x^{2} - x + 9$