# Sinusoidal Functions

## 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

Peak: highest point

Trough: lowest point

Amplitude: half the distance between the maximum and minimum values

*Always Positive

## 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
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

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

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

Practice Stating the Transformations, as well as the values of a, k, c and d for the graph below

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