Exploring
Sine Functions

Erin
Mueller

Given the following function; y=a(sin(bx+c)), the value of ÒaÓ will affect the amplitude. The
value of ÒbÓ will change the period and the value of ÒcÓ will alter the
position that our function starts at on the x-axis. The original sine function
below begins at x=0. Changing ÒcÓ will either move right or left depending on
whether ÒcÓ is positive or negative.

Above, we can see the graph of a regular sine
graph. However, watch what happens when we change certain characteristics to
the sine graph. When we change the value of ÒaÓ in y=a(sin(x)),
we will change the amplitude. The amplitude represents the maximum value for
which the sine graph will reach on the y-axis before it begins to descend. The
amplitude for the original sine function is 1.

The functions above represent the change in our ÒaÓ value. For the
purple graph, our ÒaÓ value is 2. When the ÒaÓ value becomes negative, the
graph is flipped about the y-axis. We can also see that the amplitude is 2 when
our ÒaÓ value is positive 2 or negative 2. The original sine graph is blue.
From this point, we can see the changes that our ÒaÓ value creates. Below, we
can see what happens as ÒaÓ is increased and decreased.

When we alter the value of ÒbÓ in our y=a(sin(bx+c)) function, the period changes. The period of the
original sine function is 2. This means that after x=2, the graph will begin to
repeat itself. From the graphs below, we can observe what happens when the
value of ÒbÓ is increased and decreased.

Now we know that y=sin(2x) (red graph)
essentially cuts the period of the function in half. The period is now instead of 2. Since is approximately 3.14É, this
corresponds to the point in the graph (red) above at which our graph starts to
repeat. From this, we can generalize that the period is equal to . The blue graph above simply represents the fact that when
the value of ÒbÓ is negative, our function is reflected across the x-axis. From
the video below, we can see the effects that "b" has on the sine function.

The ÒcÓ value in y=a(sin(bx+c)) represents the starting x-value for the graph. If
our ÒcÓ value is increased, the graph will shift ÒcÓ units to the left. If our
ÒcÓ value is decreased, the graph will shift ÒcÓ units to the right. This is
shown below.

As we can see from the picture above, our purple graph represents
the original sine function, y=sin(x). The red graph
represents our sine function that has shifted 2 units to the left and our blue
graph shows the sine function shifted 2 units to the right. From the video
below, we can see the changes that occur as "c" is increased and
decreased. Notice that when "c" is positive, the function moves left
and when "c" is negative, the function moves right.