# 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

**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**

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**

***Always start with the parent function**

**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**

**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. **

**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**