If you need a review of long division, feel free to go to Tutorial Long Division. Note that this rational function is already reduced down. Applying long division to this problem we get:. Practice Problems. At the link you will find the answer as well as any steps that went into finding that answer. Practice Problem 1a: Give the domain of the given function.
Practice Problems 2a - 2b: Find the vertical and horizontal asymptotes for the given functions. Practice Problem 3a: Find the oblique asymptote for the given function. Practice Problems 4a - 4b: Sketch the graph of the given function.
Need Extra Help on these Topics? The following are webpages that can assist you in the topics that were covered on this page. All rights reserved. After completing this tutorial, you should be able to: Find the domain of a rational function.
In this tutorial we will be looking at several aspects of rational functions. The domain is the set of all input values to which the rule applies. In other words, you find the vertical asymptote by locating where the function is undefined.
You can have zero or many vertical asymptotes. Example 2 : Find the vertical asymptote of the function. Nothing is able to cancel out, so now we want to find where the denominator is equal to Example 3 : Find the vertical asymptote of the function. If there is a horizontal asymptote, it will fit into one of the two following cases:. In other words, it would be the ratio between the leading coefficient of the numerator and the leading coefficient of the denominator. You may have 0 or 1 horizontal asymptote, but no more than that.
The graph may cross the horizontal asymptote, but it levels off and approaches it as x goes to infinity. Example 4 : Find the horizontal asymptote of the function. Nothing was able to cancel out, so now we want to compare the degrees of the numerator and the denominator. What is the degree of the numerator?
If you said 1, you are correct. The leading term is 3 x and its degree is 1. What is the degree of the denominator? If you said 2, you are correct. The leading term is and its degree is 2. If you need a review of finding the degree of a polynomial, feel free to go to Tutorial 6: Polynomials. Example 5 : Find the horizontal asymptote of the function. What is the degree of the numerator that is left? The leading term is x and its degree is 1.
What is the degree of the denominator that is left? Since the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at. You may have 0 or 1 slant asymptote, but no more than that.
Example 6 : Find the oblique asymptote of the function. The answer to the long division would be. The equation for the slant asymptote is the quotient part of the answer which would be. Step 1: Reduce the rational function to lowest terms and check for any open holes in the graph. They omitted a linear term in the polynomial on top, and they put the terms in the wrong order underneath.
So, when I'm doing my long division, I'll need to be careful of the missing linear term in the numerator, and of the signs when I reverse the terms in the denominator. If you're not comfortable with the long-division part of these exercises, then go back and review now! A note for the curious regarding the horizontal and slant asymptote rules. Otherwise, continue on to the worked examples. Page 1 Page 2 Page 3 Page 4. All right reserved.
Web Design by. Skip to main content. Purplemath In the previous section, covering horizontal asymptotes, we learned how to deal with rational functions where the degree of the numerator was equal to or less than that of the denominator.
Content Continues Below. To find the slant asymptote, I'll do the long division:. Share This Page. Tips and Warnings. Related Articles. Check the numerator and denominator of your polynomial. Make sure that the degree of the numerator in other words, the highest exponent in the numerator is greater than the degree of the denominator. Therefore, you can find the slant asymptote. The graph of this polynomial is shown in the picture. Create a long division problem. Place the numerator the dividend inside the division box, and place the denominator the divisor on the outside.
Find the first factor. Look for a factor that, when multiplied by the highest degree term in the denominator, will result in the same term as the highest degree term of the dividend.
Write that factor above the division box. Write the x above the division box. Find the product of the factor and the whole divisor. Multiply to get your product, and write it beneath the dividend. Write it under the dividend, as shown. Take the lower expression under the division box and subtract it from the upper expression.
Draw a line and note the result of your subtraction underneath it. Continue dividing. Repeat these steps, using the result of your subtraction problem as your new dividend. Write the product of the factor and the divisor beneath the dividend, and subtract again, as shown.
Stop when you get an equation of a line. You do not have to perform the long division all the way to the end. In the example above, you can now stop. Draw the line alongside the graph of the polynomial.
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