Graphing Rational Functions Steps End ShowEnd Show Slide #1 NextNext Step #1 Factor both numerator and denominator, but don’t reduce the fraction yet. Example

Graphing Rational Functions Steps PreviousPreviousSlide #2 NextNext Step #2 Note the domain restrictions. For rational functions, x can not be any number that makes the denominator 0. Example

Graphing Rational Functions Steps PreviousPreviousSlide #3 NextNext Step #3 Reduce the fraction. Example

Graphing Rational Functions Steps PreviousPreviousSlide #4 NextNext Step #4 Find the vertical asymptotes (V.A.). The V.A. will be where the denominator of the reduced form is 0. Remember to give the V.A. as the full equation of the line. Notice that you are using the reduced form from step #3 not the factored form from step #1. Thus, the numbers for the V.A. will be among the numbers in the domain restriction in step #2, but they won't always be all the numbers. Example

Graphing Rational Functions Steps PreviousPreviousSlide #5 NextNext Step #5 If there are any numbers that are in the domain restriction but are not in the V.A., there will be a hole in the graph at those numbers. To find the y-coordinate of the hole plug the number into the reduced form. Example

Graphing Rational Functions Steps PreviousPreviousSlide #6 NextNext Step #6 Next find the horizontal asymptote (H.A.) or the oblique asymptote (O.A.) and the intersections with the H.A. or O.A.

Graphing Rational Functions Steps PreviousPreviousSlide #7 Step #6.1 First does the numerator or the denominator have the larger degree. Choose the NEXT under the case you want to see. Case A Denominator has larger degree Example NEXT Case B Both degrees are the same Example NEXT Case C or D Numerator has larger degree Example NEXT

Graphing Rational Functions Steps PreviousPreviousSlide #8 NextNext Step #6.2 Case A When the denominator has the larger degree, y=0 is always the H.A. and there is no O.A. Example

Graphing Rational Functions Steps PreviousPreviousSlide #9 NextNext Step #6.3 Case A Since the H.A. is y=0 and y=0 is the x-axis, the intersections with the H.A. are the x-intercepts which will be done in a later step.

Graphing Rational Functions Steps PreviousPreviousSlide #10 NextNext Step #6.2 Case B When the degrees are the same the H.A. is y=( the ratio of the leading coefficients). Example

Graphing Rational Functions Steps PreviousPreviousSlide #11 NextNext Step #6.3 Case B To find the intersections with the H.A. set the reduced form equal to the H.A. and solve for x. The y-coordinate will be the same number as the H.A. Example

Graphing Rational Functions Steps PreviousPreviousSlide #12 Step #6.1.1 Case C or D Is the degree of the numerator exactly 1 more than the degree of the denominator? Case C Yes, exactly one more. Example NEXT Case D No, 2 or more. Example NEXT

Graphing Rational Functions Steps PreviousPreviousSlide #13 NextNext Step #6.2 Case C Divide out the fraction to get the form The O.A. is y=mx+b. Example

Graphing Rational Functions Steps PreviousPreviousSlide #14 NextNext Step #6.3 Case C To find the intersection with the O.A., set the function equal to the O.A. and solve for x. This is equivalent to setting the remainder to 0. Example

Graphing Rational Functions Steps PreviousPreviousSlide #15 Step #6.2 Case D In this case there is neither a H.A. nor an O.A. All that you are required to know is that the end behavior of the graph is to either turn up or turn down at each end. However, there is an optional step from College Algebra that can help determine the end behavior. Skip Optional Steps Go to Optional Steps

Graphing Rational Functions Steps PreviousPreviousSlide #16 NextNext Step #6.3 Case D Optional Step: Take the non-factored form and drop off all but the leading terms in the numerator and denominator. Then reduce to get a monomial. Note in the example below I didn't use an = when I dropped off the terms since the 2 expressions are not equal. The end behavior of the f(x) is the same as the end behavior of the monomial. Example

Graphing Rational Functions Steps Previous SlideStep #6Previous SlideStep #6Slide #17 NextNext Step #7 Find x-intercept by finding where the numerator is 0 which would be where each factor of the numerator is 0. Example

Graphing Rational Functions Steps PreviousPreviousSlide #18 NextNext Step #8 Find the y-intercept by plugging in 0 for x. Example

Graphing Rational Functions Steps PreviousPreviousSlide #19 NextNext Step #9 If you haven't done so already plot what you know about the graph from previous steps. Then plot any additional points, by choosing values of x where either 1) there is a section of the graph with no points plotted already. or 2) you are unsure about where the graph is.

Graphing Rational Functions Steps PreviousPreviousSlide #20 Restart End ShowRestart End Show Step #10 Finally, draw the graph using the information from previous steps.