# Find range of a function given the domain??

## I have the following trigonometric curves with a given domain. How do you work out the range? Domain $0 \le x \le 8$ Functions: $y = a \sin \pi x$ $y = b \cos \pi x$

Jun 25, 2018

The ranges are $\left[- a , a\right]$ and $\left[- b , b\right]$.

#### Explanation:

Your functions are a slightly modified version of the standard trigonometric functions $\sin \left(x\right)$ and $\cos \left(x\right)$.

The most general case is $A \sin \left(\omega x + \phi\right) + b$, where

• $A$ influences the amplitude. The standard sine function has amplitude $1$, i.e. has range $\left[- 1 , 1\right]$. Any modified version has amplitude $A$, i.e. ranges from $- A$ to $A$.
• $\omega$ influences the period, given the formula $T = \frac{2 \pi}{\omega}$
• $\phi$ and $b$ are shift: $\phi$ translates the function horizontally, $b$ represents a vertical translation.

In your case, the period of the function is $2$, since $\omega = \pi$ and thus

$T = \frac{2 \pi}{\omega} = \frac{2 \pi}{\pi} = 2$

for both functions. This means that the domain $\left[0 , 8\right]$ spans four periodicity of the function, and this means that the minimum and maximum of the functions are reached four times each. And given the first point, you know that the range of the first function is $\left[- a , a\right]$, while the range of the second is $\left[- b , b\right]$.