1. 1 Example 2 Sketch the graph of the function Solution Observe that g is an even function, and hence its graph is symmetric with respect to the y-axis. I. Intercepts The x-intercepts occur when 0 = x 2 -9 = (x+3)(x-3), i.e. when x=-3 or x=3. The y-intercept occurs at II. Asymptotes Vertical asymptotes occur where the denominator of g(x) is zero: 0 = x 2 -4 = (x+2)(x-2), i.e. when x=-2 and x=2. The horizontal asymptote on the right is given by: The horizontal asymptote on the left is given by: Thus g has the horizontal asymptote y=1 on both the left and the right.

2. 2 III. First Derivative By the quotient rule, the derivative of is: Since the denominator of g / (x) is always positive, g / (x) has the same sign as its numerator 10x. Hence g / (x) is positive for x>0 while g / (x) is negative for x 0 while g is decreasing for x<0. We depict this information on a number line. g only has one critical point: x=0 where g / (x) = 0 because the numerator of g / (x) vanishes. By the First Derivative Test, x=0 is a local minimum. Note that x=-2 and x=2 are not critical points of g because g has vertical asymptotes at these numbers and they are not in the domain of g.

3. 3 IV. Vertical Tangents and Cusps g has neither vertical tangents nor cusps. V. Concavity and Inflection Points By the quotient rule, the derivative of is: Observe that the numerator of g // (x) is always negative. Hence g // (x) is positive for –2 2. Therefore the graph of g is concave up for –2 2. Note that g has vertical asymptotes x=-2 and x=2, and these numbers are not in the domain of g. Hence g has no inflection points.

4. 4 VI. Sketch of the graph We summarize our conclusions and sketch the graph of g. x-intercepts: -3, +3 y-intercept: 9/4 vertical asymptotes: x=-2 and x=+2 horizontal asymptote: y=1 on the left and right g is an even function increasing: 0<x decreasing: x<0 local min: x=0 concave up: -2 2 no inflection points