Graphing Rational Functions Example #7 End ShowEnd Show Slide #1 NextNext We want to graph this rational function showing all relevant characteristics.

Graphing Rational Functions Example #7 PreviousPreviousSlide #2 NextNext First we must factor both numerator and denominator, but don’t reduce the fraction yet. Numerator: Factor out the negative. Then factor to 2 binomials. Denominator: Prime

Graphing Rational Functions Example #7 PreviousPreviousSlide #3 NextNext Note the domain restrictions, where the denominator is 0.

Graphing Rational Functions Example #7 PreviousPreviousSlide #4 NextNext Now reduce the fraction. In this case, there isn't a common factor. Thus, it doesn't reduce.

Graphing Rational Functions Example #7 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. as the full equation of the line and to graph it as a dashed line.

Graphing Rational Functions Example #7 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.

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

Graphing Rational Functions Example #7 PreviousPreviousSlide #8 NextNext 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. Also, since the denominator is a monomial, I'll divide each term in the numerator by the x.

Graphing Rational Functions Example #7 PreviousPreviousSlide #9 NextNext The O.A. will be y=(the polynomial part of the division).

Graphing Rational Functions Example #7 PreviousPreviousSlide #10 NextNext Now we need to find the intersections between the graph of f(x) and the O.A. The more formal way to find this is to set the function equal to the O.A., but you see this reduces to when the remainder is 0. Since, 4 is never 0, there are no intersections w/ the O.A.

Graphing Rational Functions Example #7 PreviousPreviousSlide #11 NextNext We find the x-intercepts by solving when the function is 0, which would be when the numerator is 0. Thus, when 3x-4=0 and x+1=0.

Graphing Rational Functions Example #7 PreviousPreviousSlide #12 NextNext Now find the y-intercept by plugging in 0 for x. In this case, there wouldn't be a y-intercept since the function is undefined at x=0.

Graphing Rational Functions Example #7 PreviousPreviousSlide #13 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.

Graphing Rational Functions Example #7 PreviousPreviousSlide #14 NextNext Finally draw in the curve. Let's start on the interval for x<0, the graph has to pass through the point (-1,0) and approach both asymptotes without crossing the O.A. or the x- axis again.

Graphing Rational Functions Example #7 PreviousPreviousSlide #15 NextNext For the interval for x>0, the graph has to pass through the point (4/3,0) and approach both asymptotes without crossing the O.A. or the x-axis again.

Graphing Rational Functions Example #7 PreviousPreviousSlide #16 End ShowEnd Show This finishes the graph.