Sinusoidal Functions
A sinusoidal function is of the form = sin
+ or = cos
+ , where A =
amplitude, B = horizontal stretch factor and D = vertical shift. Sometimes the letter M is used in place of
D. (For this set we are ignoring horizontal/phase shifts)
In order to handle these functions, you should know what the basic f(x) = sin x and f(x) = cos x functions
look like:
Part I: Graphing by hand.
Example: Sketch = −2 sin2 + 3.
Remember “MAP”: Midline, Amplitude and Period. First, identify the midline and amplitude, and lightly
sketch these “guidelines” on your graph. In this example, midline = 3 and amplitude = 2 (ignore the
leading negative for now). See figure 1.
Second, determine the period. The “2” coefficient of the x variable will give us the period, using the
formula = In this case, the period is = . Find π on the x-axis. Then subdivide this interval into 4
equal subdivisions, and lightly sketch in vertical guidelines, as shown in figure 2.
Third, this is a sine graph with a leading negative. The sine function “starts” at the midline, but the leading
negative causes it to reflect across the midline. In this case, it will go “down” from the midline first, instead
of its normal “up”. Using your guidelines, you should be able to sketch in a smooth curve: see figure 3.
For the final graph, see figure 4.
Try these. Follow the steps above and do by hand. (None of these contain horizontal shifts)
1.
2.
3.
4.
= 3 sin + 1
= −2 cos2
ℎ = −3 sin + 4
= 5 cos + 2
(Answers on the last sheet)
Part II: Determining a function from a graph.
Suppose the following graph is given:
It is easy to identify the midline and the amplitude: the midline is −1 (take the average of the lowest and
highest values), and the amplitude is 4. Now we note where it “comes off” the y-axis: it starts from the
high guideline position, which suggests the cosine function (if it came from the lower guideline, we'd use
cosine with a leading negative). So now we have = −4 cos
− 1. Now we need to find B. Since
the period is 2, we have = = . The function that generated this graph is = −4 cos − 1
For practice, see the next sheet:
Determine the correct sinusoidal function for these graphs.
(Hint on #2: how many periods do you see?)
Part III: Creating models from word problems.
When creating a model, it may help to sketch the graph first using the techniques in part I, then use that
information to create the model function. Consider this example:
The rabbit population in a field fluctuates with the seasons. In January, the cold weather and lack of food reduces the
population to 500. In July, the population rises to its high of 800. This cycle repeats itself. Determine a model.
Solution: For the sake of ease, let t = 0 represent January, and t = 6 represent July (count the months off
on your fingers; you'll see this makes sense. Also, by starting time t at t = 0, we avoid the nastiness of a
horizontal/phase shift). On a graph, plot the points (0,500) and (6,800). You should also plot (12,500) to
represent one full cycle of the rabbit population. See figure 1 below.
From this meager set of points, we can surmise the midline as being halfway between the low and high
points of the graph, in other words, the average of 500 and 800. Hence the midline is 650. Therefore, the
amplitude is 150. The period is 12, so = = . Now it's just a matter to determine whether we use
!
“sin” or “cos”. Since the placement of our three points suggests the graph comes off the y-axis at the
lower guideline, we'll use “cos” with a leading negative (see figure 2). Therefore, the model for this
situation is = −150 cos + 650.
!
Graphs are on the next page.
Try these examples:
1. The tides at a beach are cyclical. At midnight, low tide is 4 feet, while at noon, high tide is 10 feet.
Determine a model for these tides.
2. An electrical current alternates between 60 V and 120 V and back every 0.2 seconds. Assume the initial
voltage was 120 V. Determine the model.
3. A ferris wheel is 50 feet in diameter, with the center 60 feet above the ground. You enter from a
platform at the 3 o'clock position. It takes 80 seconds for the ferris wheel to make one revolution
clockwise. Find the model that gives your height above the ground at time t (t=0 when you entered).
Answers.
Part I:
Part II: 1 % = 2 sin + 3
2 % = −3 cos2 + 1
3 % = −4 sin Part III:
(1): Let the points be (0,4) and (12,10) and (24,4). The midline is the average of 4 and 10, which is 7, the
amplitude is 3. The graph starts low, so use negative cosine. The period is 24, so = & = . Therefore,
= −3 cos + 7
(2): Use points (0,120), (0.1, 60), (0.2, 120). The midline is the average of 60 and 120, which is 90, the
amplitude is 30. The graph starts “high” (top guideline), so use cosine. The period is 0.2, so =
=
(.
10. The model is = 30 cos10 + 90.
(3): Your points are (0,35), (20,10), (40,35), (60,60), (80,35). The midline is 35, amplitude is 25, and the
period 80, which gives = *( = &(. The graph starts at the midline but goes down first, so we use
negative sine. The model is = −25 sin + 35.
&(
By Scott Surgent ([email protected])
Updated 4-14-11. Notify me of any errors.