Sinusoidal Functions
By Caitlyn Schofield
Modelling Periodic Behaviour
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
there is a positive p value for every x value in the given function
Sinusoidal Functions
*Can be used to model trends or repetitive motions (eg. ferris wheel rotations)
E of A = 0
Amplitude = 1
Range = {y: -1≤y≤1}
Key Properties of f(x)= Sin(x) {0≤x≤360} - in degrees
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
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
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
Period = 360/|k|
E of A: y = c
Start Point of Cycle = d
*Always start with the parent function
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
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
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