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Graphing Rational Functions Example #3

Graphing Rational Functions Example #3. We want to graph this rational function showing all relevant characteristics. END SHOW Slide # 1 Next. Graphing Rational Functions Example #3. First we must factor both numerator and denominator, but don’t reduce the fraction yet.

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Graphing Rational Functions Example #3

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  1. Graphing Rational FunctionsExample #3 We want to graph this rational function showing all relevant characteristics. END SHOW Slide #1Next

  2. Graphing Rational FunctionsExample #3 First we must factor both numerator and denominator, but don’t reduce the fraction yet. Numerator: Factor out a GCF of x2. Denominator: Factor as a perfect square trinomial. Previous Slide #2Next

  3. Graphing Rational FunctionsExample #3 Note the domain restrictions, where the denominator is 0. Previous Slide #3Next

  4. Graphing Rational FunctionsExample #3 Now reduce the fraction. In this case, there isn't a common factor. Thus, it doesn't reduce. Previous Slide #4Next

  5. Graphing Rational FunctionsExample #3 Any places where the reducedform is undefined, the denominator is 0, forms a vertical asymptote. Remember to give the V. A. as the full equation of the line and to graph it as a dashed line. Previous Slide #5Next

  6. Graphing Rational FunctionsExample #3 Any values of x that are not in the domain of the function but are not a V.A. form holes in the graph. In other words, any factor that reduced completely out of the denominator would create a hole in the graph where it is 0. Since this example didn't reduce, it has no holes. Previous Slide #6Next

  7. Graphing Rational FunctionsExample #3 Next look at the degrees of both the numerator and the denominator. Because the denominator's degree,2, is exactly 1 less than the numerator's degree,3, there will be an oblique asymptote, but no horizontal asymptote. Previous Slide #7Next

  8. Graphing Rational FunctionsExample #3 To find the O.A. we must divide out the rational expression. In this case, since the fraction didn't reduce we will use the original form. Previous Slide #8Next

  9. Graphing Rational FunctionsExample #3 The O.A. will be y=(what is on top of the division). Previous Slide #9Next

  10. Graphing Rational FunctionsExample #3 Now we need to find the intersections between the graph of f(x) and the O.A. Usually the easiest way to do this is to set the remainder from the division equal to 0 and solve for x. Previous Slide #10Next

  11. Graphing Rational FunctionsExample #3 Next we need to find the y coordinate of the intersection by plugging the x we just found into the equation from the O.A. Previous Slide #11Next

  12. Graphing Rational FunctionsExample #3 We find the x-intercepts by solving when the function is 0, which would be when the numerator is 0. Thus, when x2=0 and x+4=0. Previous Slide #12Next

  13. Graphing Rational FunctionsExample #3 Now find the y-intercept by plugging in 0 for x. Previous Slide #13Next

  14. Graphing Rational FunctionsExample #3 Plot any additional points needed. In this case we don't need any other points to determine the graph. Though, you can always plot more points if you want to. Previous Slide #14Next

  15. Graphing Rational FunctionsExample #3 Finally draw in the curve. Let's start on the interval for x<-1, the graph has to pass through the point (-4,0) and approach both asymptotes. Previous Slide #15Next

  16. Graphing Rational FunctionsExample #3 For -1<x<0, the graph to go through the points (0,0) and (-2/5,8/5) and approach the V.A. The graph has to approach the V.A. going up since it can cross the x-axis between x=-1 and x=-2/5. Previous Slide #16Next

  17. Graphing Rational FunctionsExample #3 For x>0, the multiplicity of the x-intercept of 0 is 2. Which since it is even the graph must bounce off the x-axis. Then the graph must approach the O.A. Previous Slide #17Next

  18. Graphing Rational FunctionsExample #3 This finishes the graph. Previous Slide #18END SHOW

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