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Graph of Rational Functions

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Hi, and welcome. My name is Anthony Varela. And today we're going to talk about the graph of rational functions. So we're going to focus on asymptotes to the graph. So we'll talk about vertical asymptotes, horizontal asymptotes, and oblique or slant asymptotes.

So first, what is an asymptote? Well, it's a line that a curve approaches. And the distance between the curve and an asymptote approaches 0.

So here's a preview of the graphs that we're going to be looking at today. The dashed lines represents asymptotes. And we can see that they're not actually parts of the function of the curve itself, but they represent or dictate interesting behavior to the graph.

So let's talk about vertical asymptotes first. And now with any function, we cannot divide by 0. So to think about vertical asymptotes, so we're going to think what x values makes my denominator equal 0. So I'm going to take x squared minus 1 and set that equal to 0, and solve for x.

So first, I'll factor x squared minus 1. That's x plus 1 times x minus 1. And I can set each of those factors then equal to 0. So when x equals negative 1 and when x equals positive 1, my denominator equals 0.

Now a final step, we need to take these x values and make sure that our numerator is non-zero. Because if our numerator is 0 as well, we have a hole in the graph rather than a vertical asymptote. So the function is still undefined because we can't divide by 0. But it's not a vertical asymptote.

So we see that when x equals negative 1 our numerator is non-zero. And when x equals positive 1, our numerator is non-zero as well. So we have vertical asymptotes at these two x values.

So here's a sketch of our function. And we see our vertical asymptotes at x equals negative 1 and x equals positive 1. And looking at the behavior of the curve on the left and right sides of each vertical asymptote, we see that the function heads towards one of the infinities, either positive infinity or negative infinity.

But the function is not defined at x equals negative 1 or x equals positive 1. Remember, that would be division by 0.

So a vertical asymptote is a vertical line that a curve approaches from the left or the right. The curve tends towards positive or negative infinity, and its distance to the vertical line tends towards 0. So our function gets very, very close to a vertical asymptote, but isn't at the vertical asymptote.

Next, we're going to talk about horizontal asymptotes. And with horizontal asymptotes, this describes the behavior of the function as x approaches positive infinity or negative infinity. And so to find horizontal asymptotes, we're going to analyze the degrees of our polynomial in the numerator and the polynomial in the denominator.

So we would say then that the numerator has polynomial p of x with a degree n. And the denominator is a polynomial q of x with a degree m. So here we see a second-degree polynomial in the numerator and a third-degree polynomial in the denominator.

So when we have the degree of the numerator that is less than the degree of the denominator, our horizontal asymptote is the line y equals 0. And thinking about y, it's y equals 0. Because the degree in the denominator is greater, that means that as x approaches positive or negative infinity, the denominator is going to increase or decrease at a much more dramatic rate than the numerator. So when we divide then, it's going to approach 0.

So here is a sketch of this function. And we can see that on the extreme ends of the x-axis, our curve approaches 0. So a horizontal asymptote is a horizontal line that a curve approaches from above or below. The curve tends towards a constant value. And its distance towards the horizontal line tends towards 0.

Now we can also have cases where the degree of the numerator is equal to the degree of the denominator. We're still going to have a horizontal asymptote. But it's not going to be at y equals 0.

If the degrees are equal to each other, the horizontal asymptote is the line y equals a over b. So what's a and what's b? Well, a is the leading coefficient of the polynomial in the numerator. And b is the leading coefficient of the polynomial in the denominator.

So here we have a degree 2 polynomial on top and a degree 2 polynomial on the bottom. So our horizontal asymptote is going to be the line y equals a over b. So the ratio of our leading coefficients.

So that would be 2 divided by 1. So here's a sketch of the graph. And we see that our horizontal asymptote is the line y equals 2. So our function is going to approach y equals 2 on the extreme ends of our x-axis, negative infinity and positive infinity.

So with horizontal asymptotes, we look at the degree. If the degree in the numerator is less than the degree in the denominator, our horizontal asymptote is y equals 0. If the degrees are equal, the horizontal asymptote is y equals a/b, the ratio of those leading coefficients.

What if the degree in the numerator is greater than the degree in the denominator? That's where we have oblique asymptotes. So here, n is greater than m.

So here is a graph of a function that has an oblique asymptote. So here we see that it's not a vertical line. It's not a horizontal line.

In this function, the oblique asymptote is a line. So we can say it's equation is y equals mx plus b. And I Should note that this is the case when the degree in the numerator is one more than the degree in the denominator.

And oblique asymptotes are found by polynomial long division. So we can divide p of x by q of x. And we'll take just that polynomial part of the answer. So we'll drop the remainder. And that's going to be the equation of the oblique asymptote.

So an oblique asymptote is also called a slant asymptote. It's a line that a curve approaches as x tends towards positive or negative infinity, so similar to the horizontal asymptotes.

And it can be defined by y equals mx plus b. m can't be 0 because that would then just make it a horizontal line. And remember that y equals mx plus b is the equation to the oblique asymptote only when the degree in the numerator is 1 more than the degree in the denominator. And that's going to be almost all of the examples that you're likely to encounter when you're studying rational functions with oblique asymptotes.

So let's review our lesson on the graph of rational functions. We talked about vertical asymptotes. So you would set your denominator equal to 0 and solve for x. But make sure that you also confirm that your numerator is non-zero at those x values, otherwise you'll have a hole in the graph rather than a vertical asymptote.

With horizontal asymptotes, we analyze the degree of the polynomials in our numerator and denominator. And we can have an equation at y equals 0 or we can have the equation at y equals a/b, so the ratio of their leading coefficients.

And we also talked about oblique asymptotes. And we can find those through polynomial division.

Thanks for watching this tutorial on the graph of rational functions. Hope to see you next time.