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Graphing Rational Functions Example #4 End ShowEnd Show Slide #1 NextNext We want to graph this rational function showing all relevant characteristics.

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

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Graphing Rational Functions Example #4 PreviousPreviousSlide #3 NextNext Note the domain restrictions, where the denominator is 0.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #4 NextNext Now reduce the fraction. In this case, there are no common factors. So it doesn't reduce.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #5 NextNext Any places where the reduced form 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #6 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #7 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #8 NextNext 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, x 3 and x, we get x 2. So the end behavior of the graph of f(x) will be like that of y=x 2, a parabola that opens up.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #9 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #10 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #11 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #12 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #13 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #14 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #15 NextNext 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.

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Graphing Rational Functions Example #4 PreviousPreviousSlide #16 End ShowEnd Show This finishes the graph.

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