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Graphing Simple and General Rational Functions Section 9.2 and 9.3 WHAT YOU WILL LEARN: 1. How to graph basic rational (fraction) functions. 2. How to graph general (or more complex) rational functions. Oh Boy! Vocabulary Rational Function – is a function of the form: f ( x) p( x) q( x) Where p(x) and q(x) are polynomials and q(x) does not equal zero (why?). The Most Basic (or Parent) Graph The graph of like: y 1 x is a hyperbola that looks y The x-axis is a horizontal asymptote (why?) 10 The y-axis is a vertical asymptote. 5 -10 -5 5 10 x The domain and range are all non-zero real numbers. The two symmetrical parts are called branches. -5 -10 Get Your Calculator’s Ready • Graph the following: y 1 x y 2 x y 3 x y What do you notice? 1 x Some More Rules/Information All rational functions of the form: y a xh k have graphs that are hyperbolas with asymptotes at x = h and y = k. To draw the graph, figure out where the asymptotes are and then use an x/y table to get a couple of points on either side of the vertical asymptote. Example • Graph: y 2 x3 1 y 10 5 -10 -5 5 -5 -10 10 x You Try • Graph y 3 x 1 2 y 10 5 -10 -5 5 -5 -10 10 x More Information! All rational functions of the form y ax b cx d also have graphs that are hyper-bolas. The vertical asymptote occurs at the x-value that makes the denominator zero. The horizontal asymptote is the line: y a c Example • Graph y x 1 state the domain and range. 2x 4 y 10 5 -10 -5 5 -5 -10 10 x Let’s Try #34 on page 544 • Graph y x7 3x 8 y 10 5 -10 -5 5 -5 -10 10 x 9.3 Graphing General Rational Functions We are now going to graph more “complex” rational functions. They will be more of the type: p( x) somepolyno mial f ( x) q( x) anotherpol ynomial The x-intercepts of the graph of f are the real zeros of p(x). The graph of f has a vertical asymptote at each real zero of q(x) Degree of numerator less than degree of denominator, y = 0 is horizontal asymptote. Degree of numerator equal to degree of denominator, horizontal asymptote is at coefficient of leading term of numerator divided by coefficient of leading term of denominator. Degree of numerator is greater than degree of denominator, then no horizontal asymptote. Graphing Example • Graph: y 4 . State the domain and range. x 1 2 • The numerator has no zeros, so there is no xintercept. • The denominator has no real zeros so there is no vertical asymptote. • The degree of the numerator is less than the degree of the denominator so the line y = 0 is a horizontal asymptote. Graphing Example m<n • Graph: y 4 . State the domain and range. x 1 2 y 10 5 -10 -5 5 -5 -10 10 x You Try • Graph y x x 1 2 y 10 5 -10 -5 5 -5 -10 10 x Another Example m = n • Graph: y 3x 2 x 4 2 • The numerator has 0 as its only zero, so there is one x-intercept at (0,0). • The denominator can be factored to (x+2) (x-2) so the denominator has two zeros at x=2 and x=-2 are vertical asymptotes. • The degree of the numerator is the same as the degree of the denominator so there is a a 3 horizontal asymptote at y b 1 Another Example m = n • Graph: y 3x 2 x 4 y 2 10 5 -10 -5 5 -5 -10 10 x Let’s Do #28 Together • Graph x 4 2 y x 3 y 2 10 5 -10 -5 5 -5 -10 10 x A Side Note • We aren’t going to worry about the case where the degree of the numerator is greater than the degree of the denominator…we’ll see that in Trig/Pre-Calc. Homework Homework: page 543, 12, 16, 20-22 all, 24, 32, page 550, 26 and 30 (identify asymptotes)