Expressions of Rational Functions

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  • 0:02 Rational Functions
  • 1:11 Vertical Asymptotes
  • 2:19 Horizontal Asymptotes
  • 3:53 Putting It all Together
  • 5:39 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn how to write a rational function given just the horizontal and vertical asymptotes. Learn how to write the asymptotes and where to write them in your function.

Rational Functions

In this video lesson, we talk about rational functions. Recall that these are functions where you have a polynomial in both the numerator and denominator. Polynomials are our functions that are made up of terms with variables and numbers. These terms are either added or subtracted. A polynomial can have as many terms as it wants.

The degree of the polynomial is the number of the highest exponent it has. If the highest exponent is 3, then the degree of the polynomial is 3. An easy way to think about rational functions is as a division problem. We have one polynomial divided by another.

The cool thing about rational functions is that you can easily tell where the vertical and horizontal asymptotes are. Because we can easily tell where they are, we can just as easily write a rational function if we are given the vertical and horizontal asymptotes.

Let's take a look.

Vertical Asymptotes

Our vertical asymptotes are the vertical lines that our function approaches. These are defined as the points where our denominator is 0. So, all the points where the bottom polynomial equals 0 are where our vertical asymptotes are.

To find them we can solve our bottom polynomial by setting it equal to 0. We can factor the polynomial to our zeroes, or use other ways we have learned to solve it. Going in reverse, if we are given a vertical asymptote, we can simply write it as a factor of our bottom polynomial.

For example, if x = 1 is a vertical asymptote, then we can simply include the factor (x - 1) in the denominator. When x = 1, the denominator will equal 0 and give us a vertical asymptote. The rule here is if x = k is a vertical asymptote, then the rational function will have the factor (x - k) in the denominator.

Horizontal Asymptotes

Horizontal asymptotes are the horizontal lines that our function approaches. They work a little bit differently than vertical asymptotes. We will have a horizontal asymptote if the degree of the numerator polynomial is the same as the degree of the denominator polynomial.

The horizontal asymptote is y = 0 if the degree of the denominator polynomial is higher than the numerator polynomial. If the degrees are the same, then our horizontal asymptote is the fraction of the leading coefficients of the numerator and denominator.

For example, if our rational function is (3x - 2)/(4x + 1), then our horizontal asymptote is y = 3/4. Or, if our rational function is 8x/(7x^2 - 2), then our horizontal asymptote is y = 0 since the degree of the denominator polynomial has a degree higher than the numerator denominator.

Going the other way, if we are told that we have a horizontal asymptote of y = 7/8, then we know that the leading coefficients are 7 in the numerator and 8 in the denominator. The degree of both the numerator and denominator will be the same. If y = 0 is the horizontal asymptote, then we just need to make sure the degree of the numerator is less than the degree of the denominator.

Putting It All Together

Now, let's put it all together.

Write a rational function that has vertical asymptotes of x = 3 and x = 5 and a horizontal asymptote of y = 2.

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