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Graphing Rational Functions Example #4

Graphing Rational Functions Example #4. We want to graph this rational function showing all relevant characteristics. End Show Slide # 1 Next. Graphing Rational Functions Example #4. First we must factor both numerator and denominator, but don’t reduce the fraction yet.

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Graphing Rational Functions Example #4

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  1. Graphing Rational FunctionsExample #4 We want to graph this rational function showing all relevant characteristics. End Show Slide #1Next

  2. Graphing Rational FunctionsExample #4 First we must factor both numerator and denominator, but don’t reduce the fraction yet. Numerator: Factor by groups. Denominator: It's prime. Previous Slide #2Next

  3. Graphing Rational FunctionsExample #4 Note the domain restrictions, where the denominator is 0. Previous Slide #3Next

  4. Graphing Rational FunctionsExample #4 Now reduce the fraction. In this case, there are no common factors. So it doesn't reduce. Previous Slide #4Next

  5. Graphing Rational FunctionsExample #4 Any places where the reducedform is undefined, the denominator is 0, forms a vertical asymptote. Remember to give the V. A. and the full equation of the line and to graph it as a dashed line. Previous Slide #5Next

  6. Graphing Rational FunctionsExample #4 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 #4 Next look at the degrees of both the numerator and the denominator. Because the denominator's degree, 1, is less than the numerator's, 3,by more than 1, there is neither a horizontal asymptote nor an oblique asymptote. Thus, at the ends the graph will either curve up or curve down. Previous Slide #7Next

  8. Graphing Rational FunctionsExample #4 Optional step: Even though there isn't a H.A. or an O.A. we can determine the end behavior of the graph. By dividing the leading terms, x3 and x, we get x2. So the end behavior of the graph of f(x) will be like that of y=x2, a parabola that opens up. Previous Slide #8Next

  9. Graphing Rational FunctionsExample #4 We find the x-intercepts by solving when the function is 0, which would be when the numerator is 0. Thus, when x+2=0 and x-2=0. Previous Slide #9Next

  10. Graphing Rational FunctionsExample #4 Now find the y-intercept by plugging in 0 for x, but in this case that would lead to a 0 in the denominator. Thus, there can't be a y-intercept. Previous Slide #10Next

  11. Graphing Rational FunctionsExample #4 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 #11Next

  12. Graphing Rational FunctionsExample #4 Finally draw in the curve. For x<-2, we can use that the left end behavior to know the graph has to curve up to the left of x=-2. You could also plot more points to determine this. Previous Slide #12Next

  13. Graphing Rational FunctionsExample #4 For -2<x<0, we can use the multiplicity of the x-int.=-2 is even, 2, to get that the graph has to bounce off the x-axis and then curve up on the left the V.A. You could also plot more points instead of multiplicity. Previous Slide #13Next

  14. Graphing Rational FunctionsExample #4 For x>2, we can use the right end behavior to get that the graph has to curve up to the right of x=2. You could also plot more points to determine this. Previous Slide #14Next

  15. Graphing Rational FunctionsExample #4 For 0<x<2, we can that the multiplicity of x-int.=2 is odd, 1, to get that the graph has to cross the x-axis and then curve down on the right of the V.A. You could also plot more points instead of multiplicity. Previous Slide #15Next

  16. Graphing Rational FunctionsExample #4 This finishes the graph. Previous Slide #16End Show

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