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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.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
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sure. what is your question?
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Differences Between Laspeyres and Paasche Indices
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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