Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
As you delve deeper and deeper into the world of math, you will find functions that produce very interesting looking graphs. When you began, your graphs consisted of straight lines and single curves. But now you've learned about rational functions, which are functions made up of the division of two polynomials. This means that your graphs now consist of several curves. You also have unseen lines that divide your curves. These unseen lines that divide your curves are referred to asymptotes. Recall that, to find your vertical asymptotes, you find at what points your denominator equals 0. To find your horizontal asymptotes, you look at the degrees of your numerator and denominator polynomials. If the degree of your denominator polynomial is larger than the degree of your numerator polynomial, then your horizontal asymptote is y = 0. If the degrees are the same, then your horizontal asymptote is the division of the leading coefficients. For example, look at these two rational functions.
1. f (x) = x / x^2
2. f (x) = 2x^4 - 3x^2 / 3x^4 + 5x - 9
The first rational function has a horizontal asymptote of y = 0 because the degree of the denominator polynomial is greater than the degree of the numerator polynomial. Also, this function has a vertical asymptote of x = 0 because, if we set the denominator equal to 0 and solve, we will get x = 0. The second rational function has a horizontal asymptote of y = 2/3 because the degree of both the numerator and denominator polynomial are the same. We got y = 2/3 by isolating the leading coefficient of the numerator polynomial and the leading coefficient of the denominator polynomial.
We know how to find vertical and horizontal asymptotes. But what happens when the degree of our numerator polynomial is greater than the degree of our denominator polynomial by 1? For example, if you see a function like this.
y = x^2 + 3x +2 / x - 2
What happens then? We get what is called a slant asymptote, an asymptote that is neither horizontal nor vertical but slanted.
Finding the Slant Asymptote
You know from your previous understanding of graphs that a straight slanted line will have an equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The question now becomes, how do you find this slant asymptote and its equation from your rational function?
As it turns out, it is a fairly straightforward process. It does require that you know how to divide polynomials. To find a slant asymptote when the degree of the numerator polynomial is exactly one more than the degree of the denominator polynomial, you need to perform the division that is shown by the rational function. The slant asymptote is then your answer, not including the remainder if there is one. Remember that dividing polynomials is very similar to dividing large numbers. We take each term with its coefficient and variable and treat it like a place value when dividing numbers. For example, the constant is treated like our ones place, the term with the x variable is treated like the tens place, the term with the x^2 variable is treated like the hundreds place, and so on.
Let's look at a couple of examples to see this polynomial division in action.
Find the slant asymptote of this rational function.
f (x) = x^3 - 5x / x^2 + 1
First, we see that the degree of the numerator polynomial is one more than the degree of the denominator polynomial. So, to find the slant asymptote, we need to perform the division. We also see that the numerator is missing the x^2 and constant terms, so when we do the division, we will make sure to put in zeroes for those terms. The denominator is missing an x term, so we will put in a zero for that term as well. Performing the polynomial division, we get this:
x^3 + 0x^2 - 5x + 0 divided by x^2 + 0x +1
Subtract out the first result (x^3 + 0x^2 + x)
Equals -6x + 0 R
Our answer is x with a remainder of -6x. Our slant asymptote is the answer without the remainder part, so y = x is our slant asymptote.
Let's look at one more example. Find the slant asymptote of this rational function:
f (x) = 2x^3 + 4x^2 - 9 / 3 - x^2
This rational function also has the degree of the numerator polynomial being one greater than the degree of the denominator polynomial. We also notice that both the numerator and denominator polynomials are missing the x term, so when we divide to find the slant asymptote, we will need to include a 0x term. We also notice that the denominator is not written with our terms in order, so when we go ahead and divide, we will make sure to write the x^2 term first, then the x term, and then the constant. Going ahead with the division, we get this:
2x^3 + 4x^2 + 0x - 9 divided by -x^2 + 0x +3
Subtract out our first result (2x^3 + 0x^2 - 6x)
Gives us 4x^2 + 6x - 9
Subtract out our second result (4x^2 + 0x - 12)
Gives us 6x + 3
Our answer is -2x - 4 with a remainder of 6x + 3. Do you remember which part is your slant asymptote? Yes, the answer without the remainder. So the slant asymptote is y = -2x - 4.
Let's review what we've learned. Rational functions are functions made up of the division of two polynomials. When the degree of the numerator polynomial is one more than the degree of the denominator polynomial, we will have a slant asymptote, an asymptote that is slanted, in addition to any horizontal or vertical asymptotes. To find what this slant asymptote is, we perform the division that is shown in the function. Our slant asymptote is the answer without the remainder part.
After this lesson, you should be able to:
- Define rational function and slant asymptote
- Explain how to find the slant asymptote
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