__An Investigation of the Sine Function
__

__by__

**Tonya DeGeorge**

LetÕs begin with the basic sine function:

y= sinxIf we were to graph this function, we would get:

From this graph, we can see that the graph intersects the

x-axis at 0,, 2, etc. We can also see that the amplitude (height of each wave) is one and the period of the function is 2 (the amount of time it takes for the wave to complete one cycle). However, the function can change based on different parameter values.For example, we can rewrite the function

y= sinxasy=asin(bx+c), wherea,b, andcare real numbers. In this particular case,aandbare equal to one andcis equal to zero. In this investigation, we will see what happens to the sine function as we change values ofa,b, andc.

__What happens when
we change the value of a?__

In order to see the difference, using the graphing calculator function, we should plot the function

y= sinxandy=asin(bx+c) on the same graph, changing the values ofawhile keepingbequal to one andcequal to 0.Before plugging in different values of

a, we should first consider all the possible values ofa. Sinceais a real number, there are three possible range of values:acan be greater to zero (a> 0), equal to zero (a= 0), or less than zero (a< 0). LetÕs first investigate whena> 0.

__What happens when a
> 0?__

If we plug in 2 for

ainto the equationy=asin(bx+c), we gety= 2sinx, as shown below:

From this graph, we can see that when we change the

avalue to 2, the amplitude of the function increases. But does it work for any positive value ofa? If we choose a different value, such asa= 10 (y= 10sinx), we get:

It would seem that whenever we change the value of

a, the amplitude changes. However, if you take a closer look at the graph, we can see that the amplitude not only increases, but that it increases to the value ofa. Fora= 2, the amplitude increased to 2. Fora= 10, the amplitude increased to 10. Therefore, we can assume, that for any positive value ofa, the amplitude increases to that value.

__
__

__What happens when a
= 0?__

If we plug zero in for

a, we can see that the functiony=asin(bx+c) becomesy= 0. Hence, the function is no longer a sine function and has become linear instead. The functiony= 0 has a slope of zero and when graphed, it lies right on thex-axis.

__What happens when a
< 0?__

If we plug in -2 for

ainto the equationy=asin(bx+c), we get the functiony= -2sinx:

From here, we can see that the amplitude also increases to 2. Likewise, if we set

a= -10, we see that the amplitude increases to 10:

Therefore, we can conclude that the amplitude of the function increases to |

a|.

__But what is the difference between a and –a?__

Now, you may be wondering what the difference is between plugging in a positive value of

aand a negative value ofawhen the amplitude is changed to |a| in both cases. Well, letÕs compare what happens to the graph whena= 2 anda= -2:

From this graph, we can see that the sign of

achanges the graph. The purple line is the graph of the functiony= 2sinxand the green line is the graph of the functiony= -2sinx.We can see that whenais negative, it not only changes the amplitude of the function, but is also a reflection of the fucntiony= 2 sinx. We can also compare it fora= 10 anda= -10:

From here, we can see that we get the same results. (The purple line represents

y= 10sinxand the green line representsy= -10sinx).

We can view the same findings when looking at the following animation, where

avaries from -5 to 5:

__Conclusions about the value of a:__

* If *a* > 0, the amplitude of the function
changes to the value of *a*.

* If *a* < 0, the amplitude changes to the
value of |*a*| and is a reflection of
the function *y* = *a*sin*x*.

* If *a* = 0, then the function changes to a
linear function, *y* = 0.

__What happens when
we change the value of b?__

As we did for the investigation of

a, we will plot the functiony= sinxandy=asin(bx+c) on the same graph, changing the values ofbwhile keepingaequal to one andcequal to 0.Since

bis a real number, there are three possible range of values:bcan be greater to zero (b> 0), equal to zero (b= 0), or less than zero (b< 0). LetÕs first investigate whenb> 0.

__What happens when b
> 0?__

If we plug in 2 for

binto the equationy=asin(bx+c), we gety= sin(2x), as shown below:

From here, we can see that the graph of the function looks as though it has been compressed. So what exactly happened here? Well, if we take a look at the basic sine function again, we can see that the period of that function is 2.

However, looking back at the previous graph:

The period of the function

y= sin2xis now , instead of 2. In other words, the function can now fit two waves in the same amount of time as one wave in the basic sine function. Does this work if we change the value ofbto 3 (fory= sin(3x))? We get:

From here, we can see that there are now three complete waves in the same interval as one. But how can we express this in mathematical terms? So far, we have:

b= 1 ->y= sinx-> period: 2

b= 2 ->y= sin(2x)-> period: (or )

b= 3 ->y= sin(3x) -> period: (since three waves are in one period in comparison to the functiony= sinx)

Therefore, for all positive values of

b, we can conclude that the period of the functiony= sinbxwill be .

__What happens when b
= 0?__

If we plug zero in for

b, we can see that the function ofy= sin(bx) becomesy= sin (0)xwhich then becomesy= sin(0). If we evaluate this, we can see that sin (0) is equal to 0 and therefore the equation becomesy= 0. So whenb= 0, the function becomes linear.

__What happens when b
< 0?__

LetÕs start by plugging in -2 for

b(y= sin (-2x)) and see what we get:

By observing the graph, we can see that we obtain the same results: the period of the function changes. As in the case when

b= 2, we can see that the period is now . LetÕs see what happens whenb= -3 (y= sin(-3x)):

Again, we see that the period has changed.

b= 1 ->y= sinx-> period: 2

b= -2 ->y= sin(-2x)-> period: (or )

b= -3 ->y= 3sin(-3x) -> period: (since three waves are in one period in comparison to the functiony= sinx)

Therefore, for all negative values of

b, we can conclude that the period of the functiony= sin(bx)will be .

__But what is the difference between b and –b?__

Again, you may be wondering what the difference is between plugging in a positive value of

band a negative value ofbwhen the period is changed to in both cases. Well, letÕs compare what happens to the graph whenb= 2 andb= -2:

The purple represents the function

y= sin2xand the green represents the functiony= sin (-2x). From here we can see that they are reflections of each other. As in with the value ofa, we can see that a negative value ofbis the reflection ofy= sin(bx), for whenbis positive.

*b* changes the graph by the animation below (where *b* ranges from -5 to 5):

__Conclusions about the value of b:__

* If *b* > 0, the period of the function
changes to .

* If *b* < 0, the period of the function
changes to and is a reflection of the function *y* = sin(*bx)*.

* If *b* = 0, then the function changes to a
linear function, *y* = sin (0), which
then becomes *y* = 0.

__What happens when
we change the value of c?__

Again, we will plot the function

y= sinxandy=asin(bx+c) on the same graph, changing the values ofcwhile keepingaandbequal to one.Since

cis a real number, there are three possible range of values:ccan be greater to zero (c> 0), equal to zero (c= 0), or less than zero (c< 0). LetÕs first investigate whenc> 0.

__What happens when c
> 0?__

If we plug in 1 for

cinto the equationy=asin(bx+c), we gety= sin (x+ 1), as shown below:

From here, we can see that the graph seems to have shifted to the left 1 unit value, with the amplitude and the period the same.

What happens if we change the value of

cto 2 (y= sin (x+ 2))?

Again, we can see that the graph shifted to the left, but this time it shifted

twounits. Likewise, we can see that it shifts three units whenc= 3 (y= sin (x+ 3)):

Therefore, we can conclude that the function

y= sin (x+c) shifts to the leftcunits.

__What happens when c
= 0?__

If we plug in zero into the equation

y= sin (x+c), we will gety= sinx. Therefore, whenc= 0, the sine function does not shift in either direction.

__What happens when c
< 0?__

If we plug in -1 for

cinto the equationy=asin(bx+c), we gety= sin (x– 1), which is shown below:

From here, we can see that the graph seems to have shifted to the

right1 unit value, with the amplitude and the period the same.What happens if we change the value of

cto -2 (y= sin (x- 2))?

Or for

c= -3 (y= sin (x- 3))?

Therefore, we can conclude that the function

y= sin (x+c) shifts to the right |c| units.Again, we can see what happens as we vary c (ranging from -5 to 5) how the function changes in the animation below:

__Conclusions about the value of c:__

* If *c *>
0, the function shifts to the left *c*
units.

* If *c *< 0, the function shifts to the
right |*c*| units.

* If *c* = 0, the function does not shift in
either direction. (When *c* = 0, the function stays as *y* = sin*x*)

__Final Conclusion:__

From this investigation, we have seen the sine function
change depending on different values of *a*,
*b,* and *c*. LetÕs put all this
information together:

For the given function: *y*
= *a* sin (*bx* + *c*):

¯
When changing the value of *a*:

o If *a* > 0, the
amplitude of the function changes to the value of *a*.

o
If *a* < 0, the amplitude changes to the
value of |*a*| and is a reflection of
the function *y* = *a*sin*x*.

o
If *a* = 0, then the function changes to a
linear function, *y* = 0.

¯ When changing the value of *b*:

o
If *b* > 0, the period of the function
changes to .

o
If *b* < 0, the period of the function
changes to and is a reflection of the function *y* = sin(*bx)*.

o
If *b* = 0, then the function changes to a
linear function, *y* = sin (0), which
then becomes *y* = 0.

¯ When changing the value of *c*:

o If *c *> 0, the function shifts to the
left *c* units.

o If *c *< 0, the function shifts to the
right |*c*| units.

o If *c* = 0, the function does not shift in
either direction. (When *c* = 0, the function stays as *y* = sin*x*)