How to graph $\sec x/2$ by manipulating the cosine function?

$\sec(x) = 1/\cos(x)$. Usually, I graph $\sec(x)$ by graphing $\cos(x)$ first.

$\sec x/2= 1/2\cdot (1/\cos x) = (1/2)\cos x$. Following the same logic above, I graphed $2\cos(x)$. However, what I thought would be the answer (sec function based off red curve) was incorrect.

I understand that by function transformation rules, $\sec x/2$ would be horizontally compressed by a factor of 2. But my method above for finding the graph seems algebraically sound even though it gives the incorrect answer.

Can anyone figure out what might be wrong?

1 Answer

The correct method is to graph $y=\frac{1}{2}\cos(x)$ first, then follow the same procedure as you did for graphing $y=\sec x$. Why so? Because you want a pivot point about which you're flipping the graph of $\cos(x)$ and for that both functions must have the same magnitude, which they do at $x=n\pi, n\in\mathbb{Z}$.