Graphs of rational functions

As a grade 11 student, one of the subjects that you are required to enroll is General Mathematics. In this subject, graphing functions is one of the competencies that is needed to be mastered. One of the functions that will be graphed in this subject is the rational function.

In graphing rational functions manually, it is very important to learn first how to find the intercepts and the asymptotes, and to assign points on each region of the graph. This is not an easy competency to master. However, it is very important for us to learn and understand the basic steps. To review, here are the basic steps in graphing rational functions.

1)      Find the intercepts, if there are any. Remember that the y-intercept is given by (0, y). To find the y-intercept(s), the point(s) where the graph crosses the y-axis, substitute zero for x and solve for y. To find the x-intercept(s), the point(s) where the graph crosses the x-axis, substitute zero for y and solve for x. The x-intercept, also known as the zeros of the function, is given by (x, 0).

2)      Find the vertical asymptotes by setting the denominator equal to zero then solve.

3)      Find the horizontal asymptote, if it exists. Remember that:

·         If both the numerator and denominator are of the same degree, divide the numerical coefficients of the terms with the highest terms in both the numerator and denominator.

·         If the numerator has a degree less than the denominator, the x-axis (y = 0) is the horizontal asymptote.

·         If the numerator has a degree more than the denominator, there is no horizontal asymptote.

4)      Find the vertical asymptote(s) by simply setting the denominator equal to 0 and solve for x. The vertical asymptotes will divide the number line into regions. In each region graph at least one point in each region. This point will tell us whether the graph will be above or below the horizontal asymptote and if we need to we should get several points to determine the general shape of the graph.

5)      Sketch the graph.

Most of the students find this tiresome and mind-numbing. Thus, causing them to fail the subject or even dropping it. To lessen this problem, we can use online graphing calculators and apps. In my class, I prefer to use online graphing calculators which are free and easy to use, even without signing in. We have used Desmos and Meta-calculators in graphing.

For my students, graph 3 rational functions below using the online calculators we have used. Choose one rational function from each set. Be able to give your comment below following the format:

                                                Example:

Name:                                      Juan Dela Cruz

Chosen Rational Function:     f(x) = (3x+2)/(x-1)

URL of the graph:                  https://www.desmos.com/calculator/shevjuhxcy

x – intercept(s):                       (x, 0)

y – intercept(s):                       (0, y)

                  
                 Set A                                       Set B                                                     Set C