Calculus graphing problem

Question

Sketch two periods of the graph of the function: h(x)=5sec((Pi/4)(x+1)) Identify the stretching factor, period, and asymptotes.

Answer 1

This is the secant function with the following transformations:

Stretched vertically by 5 - from multiplying by 5

Stretched horizontally by Pi/4 (from factor pi/4).

Shifted to the left by 1 (from the x+1)

The vertical asymptotes appear anywhere the function is undefined. Here, if x = -1, then you area asking for where the sec = 0 or where the cos(x) = 1/0 which is undefined. Another way to think about it is to understand where the cos(x) = 0 namely at pi/2 and -pi/2 (there are an infinite amount of places). The functions transformation vertically does not affect this, but the horizontal and left shifts due.