This graph is the graph of y = csc x. The domain of this function is all real numbers except \(\displaystyle n\cdot \pi\) where n is any integer. In other words, there are vertical asymptotes at all multiples of \(\displaystyle \pi\). The range of this function is \(\displaystyle y\leq -1, y\geq 1\). The period of this function is \(\displaystyle 2\pi\).

This graph is the graph of y = cot x. The domain of this function is all real numbers except \(\displaystyle n\cdot \pi\) where n is any integer. In other words, there are vertical asymptotes at all multiples of \(\displaystyle \pi\). The range of this function is \(\displaystyle \left ( -\infty, \infty \right )\). The period of this function is \(\displaystyle \pi\).

This graph is the graph of y = sec x. The domain of this function is all real numbers except \(\displaystyle \frac{\pi}{2} + n\pi\) where n is any integer. In other words, there are vertical asymptotes at \(\displaystyle \frac{\pi}{2}\) , \(\displaystyle \frac{3\pi}2{}\), \(\displaystyle \frac{5\pi}{2}\), and so on. The range of this function is \(\displaystyle y\leq -1, y\geq 1\). The period of this function is \(\displaystyle 2\pi.\)

This graph is the graph of y = tan x. The domain of this function is all real numbers except \(\displaystyle \frac{\pi}{2}+n\pi\) where n is any integer. In other words, there are vertical asymptotes at \(\displaystyle \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}\), and so on. The range of this function is \(\displaystyle \left ( -\infty, \infty \right )\). The period of this function is \(\displaystyle \pi\).

The y-intercept of a function is found by substituting \(\displaystyle x = 0\). When we do this to each, we can determine the y-intercept. Don't forget your unit circle! 

\(\displaystyle y = sin(0) = 0\)

\(\displaystyle y = cos(0) = 1\)

\(\displaystyle y = csc(0) = undefined\)

\(\displaystyle y = tan(0)= 0\)

\(\displaystyle y = cot(0) = undefined\)

Thus, the function with a y-intercept of \(\displaystyle 1\) is \(\displaystyle y = cos(x)\). 

This graph is the graph of y = cos x. The domain of this function is all real numbers. The range of this function is \(\displaystyle -1\leq y\leq 1\). The period of this function is \(\displaystyle 2\pi\).

This graph is the graph of y = sin x. The domain of this function is all real numbers. The range of this function is \(\displaystyle -1\leq y\leq 1\). The period of this function is \(\displaystyle 2\pi\).

This is false. While the graphs of secant and cosecant functions are related, in order to turn a secant function into a cosecant function, you'd need to translate the original graph \(\displaystyle \frac{\pi}{2}\) units to the right to obtain a cosecant graph.

Because the tangent function is periodic, it intercepts the x-axis in infinitely many places. We can see several of these in the graph below:

In the photo, we can see that the function is intercepting the x-axis at \(\displaystyle x=-2\pi, -\pi, 0, \pi.\) Generalizing this, we can say that the tangent function intercepts the x-axis for \(\displaystyle x=n\pi\) for all values of n such that n is an integer.

This is true! Notice the similarity of the shape between the graphs, but that they intercept the x-axis at different spots, and their peaks and valleys are at different spots.

y=sin(x), passes through the point (0,0)

y=cos(x), passes through the point (0,1)