## Explore a*sin(bx+c)

by

B.J Kim

Explore y=a sin(bx+c)

At first, let a be variable and fix b=1, c=o.

So,we can investigate y=a sin(x)

By using Graphing Calculator, we can see the pattern.

## Observation

1. Every sine funtion is periodic, which is 2π.
2. Domain is the set of all real numbers.
3. Ranges are different respectively, which are [-|a|, |a|].

Hence, the value of a determines Max or Min.

Now, I will fix a=1 and c=0 but vary b.

## Observation

As b varies, the period of this sine function changed. So, we can say b affects the period of the sine function.

As |b| is larger, the period is shortened.

As |b| is smaller, the period is longer.

When we compared sin(-x) with sin(x), we observe that the two graphs are exactly the same except that sin (-x) is the reflection of sin(x) with respect to the x-axis.

Next, when c varies, what happens to the graph?

I will compared sin(x) with sin(x-1)

Note that the domain and range of this graph remain the same as those in our original case.

In addition, one period of sin(x-1) is also the same as one period of sin(x), and both equal to 2 pi.

However, we find that the graph of sin(x-1) as compared to sin(x) has a phase shift to the right by 1 unit.

Now we can try more values of c such as 2, -3, 4 and etc. Eventually, we feel that we can generalize these situations to the following:

We say that parameter c affects the phase shift of the sine function.

In particular, the sine graph shifts either to the right or to the left by c/b units. If -c/b>0, the shift will be to the right; if -c/b<0, the shift will be to the left.

## Generalization of y=asin(bx+c)

The domain of this sine function is the set of all real numbers

The range is [-|a|, |a|]. This means the max value is |a| and the min value is -|a|

One period equals 2pi/ |b|

the sine graph shifts either to the right or to the left by c/b units.

If -c/b>0, the shift will be to the right; if -c/b<0, the shift will be to the left.