# Graphing Trigonometric Functions with Translations and Asymptotes

## Key Questions

• The amplitude is the distance from the midline to the maximum or to the minimum (they are the same). For example, $y = \sin \left(x\right)$ has an amplitude of 1 because the midline is $y = 0$ and the max is 1.

This can be found by finding the range of the function and dividing by two. (See if you can figure out why.)

• $\tan x$, $\cot x$, $\sec x$, and $\csc x$ have vertical asymptotes.

I hope that this was helpful.

• By changing the "c" in your basic trigonometric equation.

The standard trig equation for sine is $y = a \cdot \sin \left[b \left(x - c \pi\right)\right] + d$. In this, the variable $a$ represents the amplitude. The variable $b$ represents the period ($\frac{2 \pi}{b}$ = period). Now, the variable $c$ represents what is known as the phase shift - more commonly known as a horizontal translation. You shift the graph $c \pi$ units from the original parent function, which in this case is $y = \sin x$. If $c$ is positive, shift the graph to the right $c \pi$ unites. If $c$ is negative, shift the graph to the left $c \pi$ units.

If you're wondering, $d$ represents the vertical translation.

I hope this helps, and I'f strongly suggest going to google and typing in functions like $y = \sin \left(x - 2 \pi\right)$ and comparing them to the parent function, $y = \sin x$.