Ex 3: Find the Intercepts and the End Behavior of a Polynomial Function

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 polynomial function in factored form, we should be able to recognize that if we multiply this out, notice how the leading term would be -3x to the third because we have factor of x here, and factor of x here, and a factor of 3x here, then we have a negative out in front.

And if it's helpful, we can also replace f of x with y and write this as y = -the quantity x + 1 x the quantity x 5 x the quantity 3x 2.

And now to find the intercepts of any function the process 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 start by doing this on the next slide. Let's go and use this form of the equation where we have y instead of f of x.

So, again, to find the y intercept we'll set x = 0 and solve for y. That would give us y = -the quantity 0 + 1 x the quantity 0 5 x the quantity 3 x 0 2. So this would give us negative and then we have 1 x -5 x this would be -2.

Notice how here we have an odd number of negatives, and therefore this product will be -10. Which means the y intercept = -10.

And now to find the x intercepts, we'll set y = 0 and solve for x. That would give us the equation 0 = -the quantity x + 1 x the quantity x 5 x the quantity 3x 2.

So this product on the right side will be equal to zero when either x + 1 = 0 or x 5 = 0, or 3x 2 = 0. So now we'll solve each of these for x to find the x intercepts.

Here we'll subtract 1 on both sides, x = -1. Here we will add 5 to both of sides, x = 5. Here we'll have two steps, we'll first add 2 to both sides, that would give us 3x = 2, divide both sides by 3, and so we have x = 2/3.

So notice how here we have three x intercepts. We have -1, +5, and 2/3. So going back to the previous slide, we can say the y intercept is -10, where as an ordered pair that would be the point (0, -10).

And we have three x intercepts, they are -1, +5, and 2/3. As ordered pairs the x intercept of -1 would be (-1, 0). The x intercept of 5 would be (5, 0).

And the x intercept of 2/3 would be (2/3, 0). And now to describe the end behavior, we want to describe the function value or y value as x approaches left or as x approaches negative infinity, and as x approaches right or as x approaches positive infinity.

To do this, we will take a look at the graph of the given function, but we should also be able to determine the end behavior by understanding how the graph of our given function will resemble the graph of the basic function y = x cubed.

The graph of y = x cubed would look something like this. And now if we take a look at our function again, notice how the leading term is going to be -3x cubed, followed by several other terms.

But we should be able to determine the end behavior of our function by knowing what the graph of y = -3x cubed would look like compared to the graph of y = x cubed.

Because the leading coefficient is -3, this would vertically stretch the graph of y = x cubed, and then because we have a negative here, this would reflect the function across the x axis.

So if we take the basic function y = x cubed, vertically stretch it, and reflect it across the x axis, it would look something like this. Now the graph of our function, because we have several other terms, won't look exactly like this, but the end behavior would be the same.

So notice using this function, notice as x approaches the right, or x approaches positive infinity, the function goes down without bound, and therefore f of x or y is approaching negative infinity. And as x approaches left, or as x approaches negative infinity, the function is increasing without bound, and therefore y or f of x is approaching positive infinity.

So this would be enough to give us the end behavior. Let's go ahead and take the time and graph the given function on our graphing calculator. So from the home screen we'll press y equals, clear out any old functions, and type in the new function.

Let's go ahead and start with the standard window by pressing zoom 6. Probably have to adjust this. And we will. We need to increase the y maximum and decrease the y minimum.

Notice how we can also change the x max and x min if we wish. So i'm going to go ahead and change the x minimum to -3 and the x maximum to 8. And let's go ahead and change the y minimum to, let's say, -50 and the y maximum to 200.

If this doesn't work, we'll just come back and change it again. That looks pretty good, let's go ahead and leave it like this.

Again, notice as x approaches positive infinity, or as we move right, the function values decrease without bound, and therefore f of x or y is approaching negative infinity. Let's go ahead and record that here.

And then as x approaches negative infinity, or as we move left, the graph is going up without bound, and therefore f of x or y is approaching positive infinity. So notice how the graph of the given function does verify what we found using the basic function f of x = -3x cubed.

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