Hello, math619!

Find the symmetries, horizontal asymptote, max and min and inflection points of

I've already found out the horizontal asymptote and the x and y intercepts.

If we replace with , we get: . . . . the same function

Therefore, the graph is symmetric to the -axis.

Vertical asymptotes occur where the denominator is zero.

. . Hence, and are vertical asymptotes.

For max and min, you need the derivative (quotient rule).

. .

The derivative equals 0 when its numerator is 0: .

To test it, we'll use the Second Derivative Test.

. .

Factor: .[1]

When . . . positive, concave up:

. . Therefore, a minimum at

Inflection points occur when the second derivative equals zero.

. . But we see from[1], that cannot equal 0.

Therefore, there arenoinflection points.