# What is the amplitude, period and the phase shift of y= -3cos (2pi(x)-pi)?

Nov 4, 2015

Amplitude is $3$.
Period is $1$
Phase shift is $\frac{1}{2}$

#### Explanation:

Amplitude is the maximum deviation from a neutral point.
For a function $y = \cos \left(x\right)$ it is equal to $1$ since it changes the values from minimum $- 1$ to maximum $+ 1$.

Hence, the amplitude of a function $y = A \cdot \cos \left(x\right)$ the amplitude is $| A |$ since a factor $A$ proportionally changes this deviation.

For a function y=−3cos(2pix−pi) the amplitude is equal to $3$. It deviates by $3$ from its neutral value of $0$ from its minimum of $- 3$ to a maximum of $+ 3$.

Period of a function $y = f \left(x\right)$ is a real number $a$ such that $f \left(x\right) = f \left(x + a\right)$ for any argument value $x$.

For a function $y = \cos \left(x\right)$ the period equals to $2 \pi$ because the function repeats its values if $2 \pi$ is added to an argument:
$\cos \left(x\right) = \cos \left(x + 2 \pi\right)$

If we put a multiplier in front of an argument, periodicity will change. Consider a function $y = \cos \left(p \cdot x\right)$ where $p$ - a multiplier (any real number not equal to zero).
Since $\cos \left(x\right)$ has a period $2 \pi$, $\cos \left(p \cdot x\right)$ has a period $\frac{2 \pi}{p}$ since we have to add $\frac{2 \pi}{p}$ to an argument $x$ to shift the expression inside the $\cos \left(\right)$ by $2 \pi$, which will result in the same value of a function.

Indeed, $\cos \left(p \cdot \left(x + \frac{2 \pi}{p}\right)\right) = \cos \left(p x + 2 \pi\right) = \cos \left(p x\right)$

For a function y=−3cos(2pix−pi) with $2 \pi$ multiplier at $x$ the period is $\frac{2 \pi}{2 \pi} = 1$.

Phase shift for $y = \cos \left(x\right)$ is, by definition, zero.
Phase shift for $y = \cos \left(x - b\right)$ is, by definition, $b$ since the graph of $y = \cos \left(x - b\right)$ is shifted by $b$ to the right relative to a graph of $y = \cos \left(x\right)$.

Since y=−3cos(2pix−pi)=-3cos(2pi(x-1/2)), the phase shift is $\frac{1}{2}$.

In general, for a function $y = A \cos \left(B \left(x - C\right)\right)$ (where $B \ne 0$):
amplitude is $| A |$,
period is $\frac{2 \pi}{|} B |$,
phase shift is $C$.