Ex 2: Find the Intercepts and the End Behavior of a Polynomial Function

In this example we're given a polynomial function in factored form and asked to find the y intercept, the x intercepts, which could also be referred to as the real zeros or real roots of the polynomial function, and then also asked to describe the end behavior using this notation here.

Looking at the given function, we should recognize that it will be degree 3 polynomial function, and if we want, we can replace f of x with y and write this as y = x x the quantity x 3 x the quantity x + 5.

And now the process to find the intercepts for any function is the same. To find the y intercept we set x = 0 and solve for y, and to find the x intercepts we set y = 0 and solve for x.

Let's go ahead and do this on the next slide. And, again, let's go ahead and use this form of our equation where we replaced f of x with y. So to find the y intercept, again, we'll set x = 0, so that would give us y = 0 x 0 3 x 0 + 5.

So we have y = 0 x -3 x 5, which of course is 0, so our y intercept = 0, which would be the origin. And now to find the x intercepts, we'll set y = 0 and solve for x.

So this would give us the equation 0 equals x x the quantity x 3 x the quantity x 5. Now, in order for this part to be 0, either x, the first factor must = 0, or the second factor of x 3 must = 0, or the third factor of x + 5 must = 0.

So solving for x, this one's already solved for x. So we have x = 0. Solving for x, we would add 3 to both sides. We have x = 3, subtracting 5 on both sides, we have x = -5.

So here notice how we have three x intercepts, 0, 3, and -5. So going back to the previous slide, the y intercept with 0-- as an ordered pair we could write this as (0.0, 0).

And we have three x intercepts, they were 0, 3, and -5. As ordered pairs, we could write the x intercept of 0 as (0, 0), x intercept of 3 would be the point (3, 0), and the x intercept of -5 would be the point (-5, 0).

And now to describe the end behavior, we want to describe the value of the function, or the y value, as x approaches the left, or as x approaches negative infinity, and as x approaches right, or as x approaches positive infinity.

To do this, we will look at the graph of given function, but we should also recognize that we can determine the end behavior by determining the lead term of the polynomial function.

Notice if we multiplied this out, the leading term would be x to the third. So for our function f of x the leading term of x cubed, followed by several other terms, we should be able to determine the end behavior by knowing what the graph of the basic function y = x cubed looks like.

The graph wouldn't be the same as our function, but the end behavior would be the same. So if we recognize that y = x cubed look something like this, we could use this function to determine our end behavior.

Notice as x approaches the right or as x approaches positive infinity, the function goes up without bound, and therefore y or f of x approaches positive infinity.

And as x approaches the left or x approaches negative infinity the function values go down without bound, and therefore y or f of x approaches negative infinity. So this would be our end behavior, but let's go ahead and verify it using the given function and our graphing calculator.

So let's go ahead and press y equals, clear out any old function, and type in our new function. We have x x the quantity x 3 x the quantity x + 5.

Let's go ahead and press graph. Again, notice how the graph does not resemble the basic function y = x cubed, but the end behavior is the same. Let's go and adjust the window to get a better view.

Let's increase the y max and decrease the y min. So i'm going to change the y min to, let's say -40 and the y max to 60.

So notice how as we approach the right, the graph is still increasing without bound, and therefore as x approaches positive infinity y is approaching positive infinity.

As we move to the left, or as x approaches negative infinity, the graph is going down without bound, and therefore y or f of x is approaching negative infinity.

So this graph does verify our end behavior. As x approaches positive infinity f of x approaches positive infinity. And as x approaches negative infinity f of x approaches negative infinity. I hope you found this explanation helpful..

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