# How do you find the period and the amplitude for y = (1/2)sin(x - pi)?

Jun 3, 2016

$2 \pi , \frac{1}{2}$

#### Explanation:

The standard form of the $\textcolor{b l u e}{\text{sine wave function}}$ is

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{y = a \sin \left(b x + c\right) + d} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

where amplitude = |a| , period$= \frac{2 \pi}{b}$

c is the horizontal shift and d , the vertical shift.

here a $= \frac{1}{2} , b = 1 , c = - \pi \text{ and } d = 0$

hence period$= \frac{2 \pi}{1} = 2 \pi \text{ and amplitude} = | \frac{1}{2} | = \frac{1}{2}$

Jun 3, 2016

$\textcolor{b l u e}{\text{Assumption: period is equivalent to 'pitch'}}$

Amplitude: $\text{ "-1/2" to } + \frac{1}{2}$

The period (pitch) is $2 \pi$

#### Explanation:

$\textcolor{b l u e}{\text{Amplitude}}$

The given equation starts with the basis of sine. This has a maximum and minimum values of -1 to +1.

Making this into a product of $\frac{1}{2} \times \text{sine}$ changes the amplitude to values $- \frac{1}{2} \text{ to } + \frac{1}{2}$
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Period}}$

I am interpreting this to be the equivalent of pitch. Like that of a thread. That is the distance between maximum values or any other such repeat.

This standard 'untampered with' period is $2 \pi$ radians or ${360}^{0}$ if you wish to think in degrees. The way to change this is to apply multiplication on the $x$ value in $\sin \left(x\right)$, say for example $\sin \left(3 x\right) \text{ or } \sin \left(\frac{1}{2} x\right)$.

As we do not have this structure the pitch remains at the standard $2 \pi$ radians.

Suppose we had $\sin \left(2 x\right)$ and suppose we were considering $x = 3$.

Imagine that we have two graphs. The one we are drawing and a reference one of $\sin \left(x\right)$

As we are considering $x = 3$ we turn to the reference graph. We look at $y = \sin \left(2 x\right) \text{ which is } y = \sin \left(6\right)$ and record that $y$ value.

We then turn to the graph we are drawing and plot this recorded $y$ value against $x = 3$. So in effect we have moved that point to the left from $x = 6$ to $x = 3$. In other words we have reduced the pitch (frequency increase)

So multiplying $x$ by $n > 1$ squashes the graph horizontally and likewise multiplying $x$ by $n < 1$ stretches the graph horizontally.

$\textcolor{b r o w n}{\text{We do not have a change in period hear!}}$

$\textcolor{g r e e n}{\text{The "sin(x-pi)" moves the whole plot to the right by "pi" radians}}$