Here we will examine the graphs of y=a sin(bx + c) for different values of a, b, and c.  Let us first look at the effects of changing a while b and c remain constant.






We can see that the amplitudes are changing, if we look more closely at the graphs of y= 1 sin (1x + 1) and 4 sin (1x + 1) we will observe the following.



Notice that the amplitude is greater with the larger value of a, and corresponds with its value on the y-axis.  So, what might we expect to see with negative values for a?



For negative values of a, the amplitude is negative and still corresponds with its value on the y-axis.  From these observations we can conclude that the change in a determines the type of vertical expansion of the function. 


Now let us observe the effects of changes in b on the graph.




Notice the graphs have different periods.  If we look at the graphs when a and c are constant and b=1, 4, and ½ we can take a closer look at how the periods are changing with the change in b.






The change in the period can be determined with the formula , therefore we can see why 0 < b > 1 would yield a horizontal expansion (shown when b = ½) and b > 1 would yield a horizontal compression.


Lastly, letŐs observe changes in c.






Here we can see that the graphs are translated horizontally.  If we take a closer look


We can see that the graph shifts to the left for c>0,




and shifts to the right for c<0.




Based on the observations we have made what do you think would happen if we changed the original equation to  y=a sin(bx + c) + d.  We have already demonstrated changes in amplitude, period, and horizontal expansion/compression, so we could logically assume that changes in d will result in a vertical expansion/compression.


LetŐs see.




We can see that when d>0 the graph shifts up,



and for values of d<0 the graph shifts down.


So for

y=a sin(bx + c) + d


|a| = amplitude

 = phase shift

c = horizontal expansion/compression

d = vertical expansion/compression