Share ThisHow to Graph a Rational Function When the Numerator and Denominator Have Equal Degrees
Rational functions with equal degrees in the numerator and denominator behave the way that they do because of limits. What you need to remember is that the horizontal asymptote is the quotient of the leading coefficients of the top and the bottom of the function.
Take a look at g (x) —
— which has equal degrees on the variables for each part of the fraction. Follow these simple steps to graph g (x):
Sketch the vertical asymptote(s) for g (x).
Try to find the value for x in which the function is undefined. Set the denominator of the rational function equal to zero, and solve this equation for x.
You find only one vertical asymptote at x = 4/3, which means you have only two intervals to consider: (–∞, 4/3) and (4/3, ∞).
Sketch the horizontal asymptote for g (x).
Because the numerator and denominator have an equal degree, you must divide the leading coefficients (the coefficients of the terms with the highest degrees) to find the horizontal asymptote.
You find that the horizontal asymptote is y = –2. So, you sketch a horizontal line at that position.
Plot the x- and y-intercepts for g (x).
To find the y-intercept of an equation, set x = 0. (Plug in 0 wherever you see x.).
To find the x-intercept of an equation, set y = 0.
You find that the intercepts are x = –2 and y = 3.
Use test values of your choice to determine whether the graph is above or below the horizontal asymptote.
The two intercepts are already located on the first interval and above the horizontal asymptote, so you know that the graph on that entire interval is above the horizontal asymptote. Now, choose a test value for the second interval greater than 4/3. In this example, you choose x = 2. Substituting this into the function g (x) gives you –12. You know that –12 is waaaay under –2, so you know that the graph lives under the horizontal asymptote in this second interval. The above figure shows you the complete graph of g (x).
