Today we're going to talk about graphs of rational functions. Rational functions have a distinguished feature called asymptotes. And an asymptote is just a line that the curve of the graph tends towards. The asymptote is not actually part of the graph itself, but a graphical feature that influences the shape of the graph.
So we'll do some examples looking at the graphs of rational functions and talking about different asymptotes, vertical, horizontal, and oblique. So let's look at an example of a rational function and its vertical asymptotes. So I have this rational function, y is equal to 2 over x squared plus 2x minus 3. And this is its associated graph.
I know that a function doesn't exist or is undefined if the denominator is equal to 0. So I can determine when this denominator is equal to 0 by writing it in factored form. So x squared plus 2x minus 3, I want to know when that's equal to zero.
And by writing it in factored form, I know that this is going to be x plus 3 and x minus 1. And that's because positive 3 and negative 1 added together will give me a positive 2, and positive 3 and negative 1 multiplied will give me negative 3. So if I want to find when this is equal to 0, then I know that using the zero product property, either x plus 3 equals 0 or x minus 1 equals 0.
Solving this first equation for x, I see that x is equal to negative 3. And solving my second equation for x, I see that x is equal to positive 1. So the two areas where or the two values of x where my function does not exist or is undefined is when x is equal to negative 3 and when x is equal to positive 1.
Looking at my graph, I can see those two values for x. So at x is equal to negative 3, I have a vertical asymptote. And when x is equal to positive 1, I have another vertical asymptote.
Now our asymptote is going to be a point on the graph or an imaginary line on the graph where the curve is never going cross. So my curves are going to go closer-- as they go closer and closer to negative infinity and positive infinity on my y-axis, the distance between my curve and the asymptote is going to get closer and closer to 0. So my curve and the asymptote will get closer and closer to 0 but never cross. Similar for this asymptote. This curve will keep getting closer and closer, but will never actually cross the asymptote.
And so we can see that the vertical asymptotes are always going to be at the same values where the function does not exist. And this is always going to be the case, unless that x value or values also makes the numerator equal to 0. And in that case, if the x value makes the numerator and the denominator equal to 0, there will simply be a hole on the graph where the function is undefined.
So let's look at an example of a rational function and a horizontal asymptote. I've got the rational function y is equal to 4x over x to the third and its associated graph. Horizontal asymptotes describe the end behavior of a graph, which is the nature of the curve as it's approaching negative infinity and positive infinity on the x-axis.
So we determine the horizontal asymptotes by looking at the degree in the numerator and the denominator. My numerator has a degree of 1. And my denominator has a degree of 3.
So since the denominator has a higher degree than the numerator, the overall value of the function is going to be approaching 0, leading to a horizontal asymptote of y equals 0. So we have a horizontal asymptote right over the x-axis at y is equal to 0.
So let's look at another example of a rational function with a horizontal asymptote. I've got the rational function y is equal to 4x plus 1 over 2x minus 2 and its associated graph. I see that the degree in my numerator is 1. And the degree in my denominator is also 1. So since the degrees in the numerator and the denominator are the same, we don't have an asymptote, a horizontal asymptote at y equals 0 like we did in the previous example.
So when the degree in your numerator and denominator are the same, the way that you find where your horizontal asymptote is by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. So the leading coefficient of my numerator is 4. And the leading coefficient of my denominator is 2.
So my asymptote, my horizontal asymptote is going to be at y equals 4 divided by 2, or y equals 2. So I have a horizontal asymptote that goes through 2 on my y-axis. And we can see that that matches our graph, that our curves are approaching closer and closer to our asymptote but will never actually cross it.
So finally let's look at an example of a rational function that has an oblique asymptote. So I have a rational function y is equal to x squared over 3x minus 2 and its associated graph. From our previous examples, we know that if the degree of the numerator is smaller than the degree of the denominator, then we have a horizontal asymptote at y equals 0. And we saw in our last example that if the degree of the numerator is equal to the degree of the denominator, then we have a horizontal asymptote at the value of the quotient of the leading coefficient of the numerator over the leading coefficient of the denominator.
But if the degree of the numerator is greater than the degree of the denominator, as with this example, then we don't have a horizontal or vertical asymptote. But we have something called an oblique asymptote. And an oblique asymptote is a diagonal line. It's neither perfectly horizontal or perfectly vertical.
And to find the oblique asymptote, you can use polynomial long division, dividing the numerator by the denominator. And an oblique asymptote would go right through these two curves. Again it's oblique, meaning a diagonal line. And the equation of an oblique asymptote is going to be when y is set equal to the quotient of the numerator and the denominator. And you would ignore any remainder of that quotient.
So let's go over our key points from today. The graph of a rational equation has a vertical asymptote at the x values that make the denominator 0, provided the x value does not make the numerator equal to 0 as well. The graph of a rational equation has a horizontal asymptote if the degree in the numerator is less than or equal to a degree in the denominator. The graph of a rational equation has an oblique asymptote if the degree in the numerator is 1 greater than the degree in the denominator. And asymptotes are not part of the curve itself, but are graphical features that influence the curve's behavior.
So I hope that these key points and examples helped you understand a little bit more about graphs of rational functions. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.
a line that a curve approaches; the distance between the curve and an asymptote approaches zero
a horizontal line that a curve approaches from above or below; the curve tends towards a constant value, and its distance to the horizontal line tends towards zero
also called slant asymptotes, a line that a curve approaches as x tends towards ± inifinity; it can be defined by y=mx+b, where m≠0
a vertical line that a curve approaches from the left or right; the curve tends towards positive or negative infinity, and its distance to the vertical line tends towards zero