We will use the graph below to develop a method to find equations that are represented by the given graph. We use the standard form: y=a sin(bx + c) + d.
a: amplitude This is the graph's displacement from the horizontal axis. To calculate the amplitude, you use the formula: (maximum y-value - minimun y-value)/2.
b: The coefficient of the independent variable divided into 2p gives you the period of the function. The period is the distance needed on the horizontal axis to complete one cycle. Another way of looking at the b is that b is the number of complete cycles completed in a 2p distance on the horizontal axis. So to calculate b, set 2p/b = the period. For tangent curves, you should use p/b, since the period of the tangent is p.
c: The phase shift (or horizontal shift) is represented by -c/b. Once you have found b, set -c\b = the starting point of a new cycle. Sine curves have a "i-M-i-m-i" cycle, where i = intercept, M = Maximum and m = minimum. Cosine curves have a "M-i-m-i-M" cycle. Since putting a "-" sign in front of a function, reflects the function around the x-axis, then -sin and -cos have the same cycle, only the maximum and minimum points have been switched.
d: Vertical shift up or down. The "d" represents how many units the horizontal axis has been shifted up or down.
Let's work through one equation for the graph above:
a: It is fairly easy to see that the amplitude of this graph is 3. So we can start by replacing the "a" in a sin(bx + c) + d with 3.
b: Since it takes from x = p/4 until x = 5p/4 to complete its i-M-i-m-i cycle, the period will be 5p/4 - p/4 = p. So we can now say 2p/b = p. So b = 2.
c: Sinc -c/b is the starting point of a new cycle, then we can set -c/b = p/4. Knowing that b = 2, we can solve to find that c = -p/2.
d: The x-axis is the horizontal axis, so there has been no vertical shift and d = 0
These calculations give you the equation, y = 3 sin(2x - p/2). See if you can calculate 3 more.
y = -3 sin(2x + p/2)
y = 3 cos(2x - p)
y = -3 cos(2x)
Now practice these transformations by looking at the following web site:
Now you are ready to print out a