1 Example 7 Sketch the graph of the function Solution I. Intercepts The x-intercepts occur when the numerator of r(x) is zero i.e. when x=0. The y-intercept occurs at r(0)=0. II. Asymptotes Vertical asymptotes occur where the denominator of r(x) is zero, i.e at x=1. The horizontal asymptote on the right is given by: The horizontal asymptote on the left is given by: Thus r has the x-axis as a horizontal asymptote on both the left and the right.

2 III. First Derivative By the quotient rule, the derivative of is: The denominator of r / (x) is always positive so the sign of r / (x) is determined by the sign of its numerator 1+2x. Hence r / (x) is positive for –1/2<x where r is increasing while r / (x) is negative for x<-1/2 where r is decreasing. We depict this information on a number line. r has two critical points: x=-1/2 where r / (x)=0 and x=0 where r / (x) does not exist. By the First Derivative Test, x=-1/2 is a local minimum and x=0 is not a local extremum. Note that x=1 is not a critical point of r because r has a vertical asymptote there and x=1 not in the domain of r.

3 IV. Vertical Tangents and Cusps r has a vertical tangent at x=0 because both the left and right derivatives have value + . V. Concavity and Inflection Points By the quotient rule, the derivative of is: Then r // (x)>0 for A 1 and r is concave down there. We depict this information on a number line.

4 Since the concavity of r changes from down to up at x=A and x=B, the function r has an inflection point there. Since the concavity of r changes from up to down at x=0, the function r has an inflection point there. Note that x=1 is not an inflection point because r has a vertical asymptote at x=1 and x=1 is not in the domain of r. VI. Sketch of the graph We summarize our conclusions and sketch the graph of r.

5 x-intercepts: x=0 y-intercept: y=0 vertical asymptote: x=1 horizontal asymptote: x-axis on the left and right vertical tangent: y-axis decreasing: x<-1/2 increasing: -1/2<x local min: x=-1/2 concave up: A 1 inflection point: x=A, x=0, x=B