Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 245.2-5.3 Properties and Graphs of Rational FunctionsChapter 5 Sections 2+3: Properties and Graphs of Rational FunctionsIn this section, we will…Express a rational function in simplest formFind the domain and range of a rational functionFind any horizontal, vertical and oblique (slant) asymptotes of rational functionsIdentify holes in the graph of a rational functionFind the intercepts of a rational functionGraph rational functionsSolve applications of quadratic functionsUse your knowledge of transformations to choose the graph of the equation:Base function:Transformations:A) B)C) D)5.2-5.3 Properties and Graphs of Rational Functions1( ) 12f xx= ++A rational function is the quotient (ratio) of two polynomials.( )( ) , ( ) 0( )p xR x q xq x= �Example: When are rational expressions undefined?Examples: Find the domain of each function.5.2-5.3 Properties and Graphs of Rational Functions5( )2 1g xx=-23 2( )7 6xh xx x-=- +243 5 2( )1x xf xx+ -=-5.2-5.3 Properties and Graphs of Rational FunctionsExample: Graph (from the last slide) with the following window settings: • How many branches does the graph have?• What do you notice about where the vertical asymptotes occur? 23 2( )7 6xh xx x-=- +Remember to place the numerator and denominator into parentheses.5.2-5.3 Properties and Graphs of Rational FunctionsThe vertical asymptote(s) of a rational function, which is in simplest form, will occur at the x-values where the function is undefined.Examples: Find the vertical asymptotes of the graph of each function.Remember the function is undefined when the denominator is equal to zero.2(1 ) ( )3 5 2x xf xx x- -� =+ -43 g( )1xxx-� =+5.2-5.3 Properties and Graphs of Rational Functions2 3( )xf xx+=Example: Graph in the standard viewing window. We will investigate the horizontal asymptote numerically using the TI table setup.• How many branches does the graph have?• Complete the tables using the TI table feature:• What do you notice about where the horizontal asymptote occurs? x y1101001,00010,000x y1- 10- 100- 1,000- 10,0005.2-5.3 Properties and Graphs of Rational FunctionsTo find the horizontal asymptote(s) of a rational function, we compare the degree of the numerator and the degree of the denominator of the function.• If the degree of the numerator is less than the degree of the denominator, then the graph of the rational function will have a horizontal asymptote at • If the degree of the numerator is equal to the degree of the denominator, then the graph of the rational function will have a horizontal asymptote at • If the degree of the numerator is greater than the degree of the denominator, then the graph of the rational function will have no horizontal asymptote (but the graph may have an oblique or slant asymptote)Examples: Find the horizontal asymptotes of the graph of each function.225 2( )3 1xf xx x+=+ -22 3( )7xg xx+=-35( )2 1xh xx=+0y =leading coefficient of polynomial in numeratorleading coefficient of polynomial in denominatory =5.2-5.3 Properties and Graphs of Rational FunctionsRecall that if the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptote but it may have an oblique (slant) asymptote.If the degree of the numerator is exactly one greater than the degree of the denominator, then the graph of the rational function will have an oblique (slant) asymptote at y = ax + b where ax + b is the quotient obtained when we divide the numerator by the denominator.Example: Find the oblique asymptote of the graph of the functionTo find the oblique asymptote use long division (or synthetic division if possible). Ignore the remainder22 5 1( )1x xf xx- +=+5.2-5.3 Properties and Graphs of Rational FunctionsExamples: Consider the rational function• We know that the domain cannot include• Write the function in lowest terms:• Graph•Notice that the graph of will be the same as the graph of the line with the exception of the hole that appears with an x-value ofand the coordinate of29( )3xh xx-=+29( )3xh xx-=+29( )3xh xx-=+5.2-5.3 Properties and Graphs of Rational Functions• Recall that the vertical asymptote(s) of a rational function will occur at the x-values where the rational function in simplest form is undefined. • So what happens at the x-values where the original rational function (not in simplest form) is undefined? Why holes of course!• Do you know your asymptote from a hole in your function?Example: Determine if the following rational functions have a vertical asymptote or a hole. Find the vertical asymptote or hole.22 5 2( )2x xf xx- +=-The x-values which cause the original rational function to be undefined give rise to either vertical asymptotes or holes. 212( )5x xf xx+ -=-5.2-5.3 Properties and Graphs of Rational Functions1. Write the rational expression in simplest form. (Factor the numerator and denominator and divide out any common factors.)2. Find the domain of the rational expression3. Find the coordinates of any “holes” in the graph.(The x-coordinate comes from the common factor; the y-coordinate is found by substituting the x-coordinate into the reduced form of the function.)4. Find the vertical asymptote(s), if any, by finding the zeros of the denominator (of the reduced form of the function). Sketch these using dashed lines. 5. Find the horizontal asymptote, if any, by comparing the degrees of the numerator and denominator. Sketch these using dashed lines.Guidelines for Graphing Rational Functionscontinued5.2-5.3 Properties and Graphs of Rational Functions6. Find the oblique asymptote, if any, by dividing the numerator by the denominator using long division. Write in the form of y = mx + b, where mx + b is the quotient. Ignore the remainder.6. Find and plot the y-intercept, if any, by evaluating f(0). (Substitute x=0 into expression.)7. Find and plot the x-intercept(s), if any, by finding the zeros of the numerator. (Set the numerator equal to zero and solve for x.)8. Plot 5-10 additional points, including points close to each x-intercept and vertical asymptote. 9. Use smooth curves to complete the graph. Guidelines for Graphing Rational Functions (continued…)5.2-5.3 Properties