Exploring Sine Functions
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.