Sinusoidal Functions

By Caitlyn Schofield

Modelling Periodic Behaviour

Periodic: repeats at regular intervals (y-values)


Cycle: a session of the pattern a periodic function is following


Period (P): horizontal distance of one cycle

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Peak: highest point


Trough: lowest point

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Amplitude: half the distance between the maximum and minimum values

*Always Positive

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Equations and Formulas

Periodic Function: f(x+p) = f(x)


f = function

p = period


Amplitude: y = (max value - min value)/2


Equation of the axis: y = (max value + min value)/2


*y can be negative

*A function is periodic when...


there is a positive p value for every x value in the given function

Sinusoidal Functions

Functions that can be produced by altering the sine function (with compression, shifting or stretching)


*Can be used to model trends or repetitive motions (eg. ferris wheel rotations)

State the period, equation of axis, amplitude and range of the function below
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Period = 5.2


E of A = 0


Amplitude = 1


Range = {y: -1≤y≤1}

State which are periodic functions and which aren't

Key Properties of f(x)= Sin(x) {0≤x≤360} - in degrees

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Amplitude: 1


Period: 360 degrees


Max: 1


Min: -1


Domain: {x:0≤x≤360}


Range: {y:-1≤y≤1}


*Pattern for every 90 degrees: Zero - Max - Zero - Min - Zero

Key Properties of f(x)= Cos(x) {0≤x≤360} - in degrees

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Amplitude: 1


Period: 360 degrees


Max: 1


Min: -1


Domain: {x:0≤x≤360}


Range: {y:-1≤y≤1}


*Pattern for every 90 degrees: Max - Zero - Min - Zero - Max

Transformations of Trigonometric Functions

Remember...


f (x) = a sin [ k (x - d) ] + c


a = vertical stretch when 1<a, compression when 0<a<1


*Reflection along x-axis is caused when a<0


k = horizontal stretch when 0<k<1, compression when 1<k


* Reflection along y-axis is caused when k<0


c = vertical translation up when 0<c, translation down when c<0


*c = # of units translated vertically


d = horizontal translation left when 0<d, translation right when d<0


*d = # of units translated horizontally

Effect of Transformations on Key Properties

Amplitude = |a|


Period = 360/|k|


E of A: y = c


Start Point of Cycle = d


*Always start with the parent function

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Practice Stating the Transformations, as well as the values of a, k, c and d for the graph below
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f(x) = 2 cos x + 3


a = 2

k = 1

c = 3

d =0


1. Vertical stretch by a factor of 2

2. Vertical translation up 3 units

Practice Graphing Transformations

Write out the steps, and graph the following functions...


1. y = 3 cos (x - 30) + 5

2. y = -2 sin (x - 45) +2

3. y = -1/2 cos (x + 60) -3

4. y = 1/2 sin (x - 45) +2

5. y = 2 cos (1/3(x-30)) -4


Steps:


1) -Parent function y = cos x

-Vertical stretch by a factor of 3

-Vertical translation up 5 units

-Horizontal translation right 30 units


2) -Parent function y = sin x

-Vertical stretch by a factor of 2

-Reflection along x-axis

-Vertical translation up 2 units

-Horizontal translation right 45 units


3) -Parent function y = cos x

-Vertical compression by a factor of 1/2

-Reflection along x-axis

-Vertical translation down 3 units

-Horizontal translation left 60 units


4) -Parent function y = sin x

-Vertical compression by a factor of 1/2

-Vertical translation up 2 units

-Horizontal translation right 45 units


5) -Parent function y = cos x

-Vertical stretch by a factor of 2

-Horizontal stretch by a factor of 3

-Vertical translation down 4 units

-Horizontal translation right 30 units


6) -Parent function y = sin x

-Vertical compression by a factor of 1/2

-Horizontal compression by a factor of 1/2

-Vertical translation up 3 units

-Horizontal translation left 45 units


*Use Desmos to check your graphs

Determining the Equation of a Sinusoidal Function

Given a function has a max of 8, min of -2, and a period of (60), state its equation


a = [8 - (-2)]/2

a = 5


c = [8 + (-2)]/2

c = 3


Period = 60

60 = 360/|k|

k = 6


d = 0


y = 5 cos (6x) +3


*Use cos when the graph starts at its maximum

Solving Problems Using Sinusoidal Models

Example


The temperature outside fluctuates throughout the day, depending on the hour. It reaches its hottest point of 30 degrees Celsius at 12:00 pm, and its coolest point of 10 degrees at hour 12:00 am. Determine the equation to represent this function.


Steps:


1. Graph/sketch the problem from the information given

2. Determine a, c, k and d from the max and min given

*a = (30-10)/2 = 10

*c = (30+10)/2 = 20

*k = 360/period = 12 x 2 = 24, k = 360/24 k = 15

*period = (horizontal value between the max and min) x2

*d = 0 (starting point is at zero on y-axis which equals 12:00 am)

3. State as an equation using the format f(x) = a sin (k (x - d)) + c


Answer:

y = 10 sin (15 x) + 20