1 Example 3 Sketch the graph of the function Solution Observe that h is an odd function, and its graph is symmetric with respect to the origin. I. Intercepts The x-intercepts occur when 0 = x 3, i.e. when x=0. The y-intercept occurs at h(0)=0. II. Asymptotes Vertical asymptotes occur where the denominator of h(x) is zero: 0 = x 2 -1 = (x+1)(x-1), i.e. when x=-1 and x=1. Since the degree 3 of the numerator of h is one larger than the degree 2 of the denominator, the graph of h has an oblique asymptote which we can find by long division. Hence and the graph of h has the line y=x as an oblique asymptote on both the left and the right.

2 III. First Derivative By the quotient rule, the derivative of is: Since the denominator of h / (x) is always positive, h / (x) has the same sign as its numerator. Since x 2  0, h / (x) has the same sign as Hence h / (x) is positive for while h / (x) is negative for Thus h is increasing for while h is decreasing for We depict this information on a number line. h has three critical points: at x=0, where the numerator of h / (x) vanishes. By the First Derivative Test, is a local maximum, x=0 is not a local extremum and is a local minimum. Note that x=-1 and x=1 are not critical points of h because h has vertical asymptotes at these numbers and they are not in the domain of h.

3 IV. Vertical Tangents and Cusps h has neither vertical tangents nor cusps. V. Concavity and Inflection Points By the quotient rule, the derivative of is: Observe that x 2 +3 in the numerator of h // (x) is always positive. Hence h // (x) is positive for –1 1 and negative for x 1 and concave down for x<-1 or 0<x<1. Note that h has vertical asymptotes x=-1 and x=1, and these numbers are not in the domain of h. Hence h has only one inflection point x=0 where the concavity changes from up to down.

4 VI. Sketch of the graph We summarize our conclusions and sketch the graph of h. x-intercepts: 0 y-intercept: 0 vertical asymptotes: x=-1 and x=+1 oblique asymptote: y=x on the left and right increasing: decreasing: local max: local min: concave up: -1 1 concave down: x<-1 or 0<x<1 inflection point: x=0 h is an odd function