Advanced Trigonometric Graphs
Master combined transformations of trigonometric functions, understand amplitude, period, phase shift, and vertical shift, and model real-world periodic phenomena.
The General Sinusoidal Form
Any sinusoidal function can be written in the general form:
y = a sin(b(x - c)) + d
or equivalently y = a cos(b(x - c)) + d
Visual: Anatomy of a Transformed Sine Curve
Combined Transformations
When graphing, apply transformations in this order:
- Horizontal shift by c (phase shift): shift the base graph right by c units.
- Horizontal scaling by factor 1/b: the period becomes 2π/b.
- Vertical scaling by factor a: the amplitude becomes |a|. If a < 0, reflect in the x-axis.
- Vertical shift by d: move the entire graph up (d > 0) or down (d < 0).
Common mistake: Forgetting to factor out b before identifying the phase shift. In y = 3sin(2x - π), rewrite as y = 3sin(2(x - π/2)) to see the phase shift is π/2 to the right, not π.
Modelling Periodic Phenomena
Many real-world situations exhibit periodic (repeating) behaviour that can be modelled with trigonometric functions:
Tides
Water height oscillates between high and low tide with a period of approximately 12.4 hours.
Temperature
Daily temperature follows a roughly sinusoidal pattern with a 24-hour period.
Ferris Wheels
Height above ground varies sinusoidally with the rotation period.
Sound Waves
Pressure variations in sound are modelled by sine and cosine functions.
Strategy for modelling: Identify the maximum, minimum, and period from the data. Then: amplitude = (max - min)/2, centre line d = (max + min)/2, b = 2π/period.
Key Vocabulary
Amplitude
The maximum displacement from the centre line; equal to |a| in y = a sin(b(x-c)) + d.
Period
The horizontal length of one complete cycle; equal to 2π/|b|.
Phase Shift
The horizontal translation of the graph; equal to c in y = a sin(b(x - c)) + d.
Centre Line
The horizontal line y = d about which the curve oscillates.
Worked Examples
State the amplitude, period, phase shift, and centre line of y = 3 sin(2(x - π/4)) + 1.
Amplitude: |a| = |3| = 3
Period: 2π/|b| = 2π/2 = π
Phase shift: c = π/4 to the right
Centre line: y = d = 1
Find the equation of a sine function with amplitude 5, period 8, and centre line y = 2.
Step 1: a = 5 (amplitude), d = 2 (centre line)
Step 2: Period = 2π/b = 8, so b = 2π/8 = π/4
Step 3: No phase shift mentioned, so c = 0
Answer: y = 5 sin(πx/4) + 2
The depth of water in a harbour varies between 2 m and 10 m. The period is 12 hours, with a maximum at t = 3. Write a cosine model.
Step 1: Amplitude = (10 - 2)/2 = 4, centre line d = (10 + 2)/2 = 6
Step 2: b = 2π/12 = π/6
Step 3: Cosine has a max at x = 0, but our max is at t = 3, so c = 3
Answer: D(t) = 4 cos(π(t - 3)/6) + 6
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the period of y = 2 sin(3x)?
Question 2
For y = -4 cos(x) + 3, what are the maximum and minimum values?
Question 3
The function y = sin(2x - π) can be rewritten as y = sin(2(x - c)). What is c?
Question 4
A periodic phenomenon has a maximum of 20 and a minimum of 4. What is the amplitude?
Question 5
Which transformation does the negative sign cause in y = -sin(x)?
Key Concepts Summary
- ● The general form y = a sin(b(x - c)) + d captures all sinusoidal transformations.
- ● Amplitude = |a|, Period = 2π/|b|, Phase shift = c, Centre line = d.
- ● Always factor out b before identifying the phase shift.
- ● For modelling: use the data to find amplitude = (max - min)/2, d = (max + min)/2, and b = 2π/period.
- ● A negative value of a reflects the graph in the x-axis.