Graphing Rational Functions Section 9-3

Terms To Know Rational Function- an equation of the form where p(x) and q(x) are polynomial functions and . Ex. Continuity- a graph of a function that can be traced with a pencil that never leaves the paper. Asymptote- a line that a graph approaches but never crosses. Point Discontinuity- If the original function is undefined for x=a but the related rational expression of the function in simplest for is defined for x=a, then there is a hole in the graph at x=a.

Background Knowledge The denominator can’t equal zero because division by zero is undefined. Not all rational functions are traceable. The graphs of rational functions may have breaks in continuity. Breaks in continuity can appear as a vertical asymptote or as a point discontinuity.

Vertical Asymptotes Property Vertical Asymptote Words If the rational expression of a function is written in simplest form and the function is undefined for x=a, then x=a is a vertical asymptote Example For X=3 is a vertical asymptote. Model

Point Discontinuity Property Point Discontinuity Words If the original function is undefined for x=a but the rational expression of the function in simplest form is defined for x=a, then there is a hole in the graph at x=a. Example Can be simplified to f(x)=x-1. So, x=-2 represents a hole in the graph. Model

X=5 is a vertical asymptote, and x=1 represents a hole in the graph. Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of 2 2 First factor the numerator and denominator of the rational expression. 2 2 The function is undefined for x=1 and x=5. Since… 1 1 X=5 is a vertical asymptote, and x=1 represents a hole in the graph.

Graph The function is undefined for x=2. Since is in simplest form, x=2 is a vertical asymptote. Draw a vertical asymptote and make a table of values. Plot the points and draw the graph. x f(x) -50 0.96154 -30 0.9375 -20 0.90909 -10 0.83333 -2 0.5 -1 0.33333 1 3 4 2 5 1.6667 10 1.25 20 1.1111 30 1.0714 50 1.0417 *as |x| increases, it appears that the y values of the function get closer and closer to 1. The line with the equation f(x)=1 is a horizontal asymptote of the function.

2 Graph 2 Notice that or x-3. 2 Therefore, the graph of is the graph of f(x)=x-3 with a hole at x=-3

Taylor did not include a slide for homework. What were you thinking? Don’t you want you classmates to learn more. Without the extra practice they may forget everything you showed them. Do not worry class, I found some homework for you. Page 488 #1-31 odd, 40-42 all