Question

graph the following function: y = -2 - \frac{3}{2}\sec(x - 2\pi)
step 1 of 2: identify the shape of the more basic function that has been shifted, reflected, stretched or compressed.

Explanation:

Step1: Recall secant - function properties

The basic secant function is $y = \sec(x)$. The graph of $y=\sec(x)$ has vertical asymptotes at $x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}$ and has U - shaped curves between the asymptotes.

Step2: Analyze the given function

The function $y=-2-\frac{3}{2}\sec(x - 2\pi)$ is a transformation of the basic secant function $y = \sec(x)$. The $x-2\pi$ inside the secant function represents a horizontal shift of $2\pi$ units to the right (but since the period of $y = \sec(x)$ is $2\pi$, it is equivalent to no horizontal shift in terms of the graph's appearance). The coefficient $\frac{3}{2}$ stretches the graph vertically by a factor of $\frac{3}{2}$, and the negative sign in front of the secant part reflects the graph about the $x$ - axis. The $- 2$ shifts the graph vertically down by 2 units. But the basic shape is that of a secant function.

Answer:

The bottom - right graph (the graph with U - shaped curves and vertical asymptotes) represents the basic shape of the secant function from which $y=-2-\frac{3}{2}\sec(x - 2\pi)$ is transformed.