A sinusoid is f(x) = a sin(b(x − c)) + d. Each letter controls exactly one feature.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Amplitude and midline
The amplitude is the size of a, half the distance from peak to trough. The midline is y = d.
Period
The period is 2π ÷ b in radians, or 360 ÷ b in degrees. A larger b squeezes more cycles into the same space.
Phase shift
The shift is c, once the inside is factored. sin(2x − π) is sin(2(x − π/2)), a shift of π/2 and not π.
Building a model
Midline is the average of max and min, amplitude is half their gap, and b comes from the observed period.
Sine or cosine
Start with cosine if the data begins at a maximum and sine if it begins at the midline rising. Choosing well removes the phase shift.
Four parameters
In y = A sin(B(x − C)) + D: A is amplitude, 2π/B is the period, C the horizontal shift, D the midline. Four numbers determine the curve completely, and each is separately readable from a graph.
Amplitude is half the swing
Amplitude measures from the midline to a peak, not trough to peak. It is half the difference between maximum and minimum, and the midline is their average. Both are extracted the same way from data.
Factor before reading the shift
sin(2x − π) must be read as sin(2(x − π/2)), giving a shift of π/2 rather than π. Failing to factor out the B is the standard error and it scales the shift by the wrong amount.
Fitting to periodic data
Midline from the average of extremes, amplitude from half their difference, period from the cycle length, shift from where a peak occurs. Four readings and the model is written, which is a standard exam task.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Kofi Model a Ferris Wheel
A wheel carries riders between 2 m and 42 m, once every 8 minutes, starting at the bottom.
- Step 1
The midline is the average of the heights: (42 + 2) ÷ 2 = 22.
Step 4: Your Turn
Practice makes it stick.
The Amplitude
Problem 1 of 2
Max 42 and min 2. What is the amplitude?
The Midline
Problem 2 of 2
Same values. What is the midline?
Read the Wave
1 of 8
y = 7 sin x. What is the amplitude?
2 of 8
y = sin x + 4. What is the midline value?
3 of 8
y = sin(2x) in degrees. What is the period?
4 of 8
y = sin(4x) in degrees. What is the period?
5 of 8
y = 3 sin x + 5. What is the maximum value?
6 of 8
Max 20 and min 4. What is the amplitude?
7 of 8
Match each parameter with the feature it controls.
Tap a card on the left to start.
8 of 8
y = 2 sin x − 1. What is the minimum value?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Max 30 and min 10. What is the midline value?
Question 2 of 2
Why factor the inside before reading a phase shift?
What You Learned
- A sinusoid is a sin(b(x − c)) + d, with each letter controlling one feature.
- Amplitude is half the peak-to-trough gap and the midline is their average.
- The period is 2π ÷ b, and the phase shift needs the inside factored first.