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In this section, you will:
  • Determine the amplitude and period of sinusoidal functions.
  • Model equations and graph sinusoidal functions.
  • Model periodic behavior.
  • Model harmonic motion functions.
Photo of the top part of a clock.
The hands on a clock are periodic: they repeat positions every twelve hours. (credit: “zoutedrop”/Flickr)

Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.

Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.

So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.

Determining the amplitude and period of a sinusoidal function

Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function    . The amplitude    of a sinusoidal function is the distance from the midline to the maximum value, or from the midline to the minimum value. The midline    is the average value. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles. Recall from Graphs of the Sine and Cosine Functions that the period    of the sine function and the cosine function is   2 π .   In other words, for any value of   x ,

sin ( x ± 2 π k ) = sin x     and     cos ( x ± 2 π k ) = cos x     where  k  is an integer

Standard form of sinusoidal equations

The general forms of a sinusoidal equation are given as

y = A sin ( B t C ) + D  or  y = A cos ( B t C ) + D

where amplitude = | A | , B is related to period such that the  period = 2 π B , C   is the phase shift such that   C B   denotes the horizontal shift, and   D   represents the vertical shift from the graph’s parent graph.

Note that the models are sometimes written as   y = a sin ( ω t ± C ) + D   or   y = a cos ( ω t ± C ) + D , and period is given as   2 π ω .

The difference between the sine and the cosine graphs is that the sine graph begins with the average value of the function and the cosine graph begins with the maximum or minimum value of the function.

Showing how the properties of a trigonometric function can transform a graph

Show the transformation of the graph of   y = sin x   into the graph of   y = 2 sin ( 4 x π 2 ) + 2.

Consider the series of graphs in [link] and the way each change to the equation changes the image.

Five graphs, side by side, each showing a manipulation to the former. (A) has y=sin(x). (B) has y=2sin(x), which has double the amplitude. (C) has y=2sin(4x), which quadrupled the frequency (or quartered the period). (D) has y=2sin(4x-pi/2), which shifted it on the x-axis by pi/2. (E) has y=2sin(4x-pi/2) + 2, which shifted it on the y-axis by 2.
(a) The basic graph of   y = sin x   (b) Changing the amplitude from 1 to 2 generates the graph of   y = 2 sin x .   (c) The period of the sine function changes with the value of   B , such that  period = 2 π B . Here we have   B = 4 , which translates to a period of   π 2 . The graph completes one full cycle in   π 2   units. (d) The graph displays a horizontal shift equal to   C B , or   π 2 4 = π 8 . (e) Finally, the graph is shifted vertically by the value of   D . In this case, the graph is shifted up by 2 units.
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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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