A sinusoid is y = a sin(b(x − c)) + d, and 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 peak-to-trough distance. The midline is y = d.
Period
The period is 360 ÷ b in degrees, or 2π ÷ b in radians. A larger b squeezes in more cycles.
Horizontal changes run backwards
Doubling b halves the period, and subtracting inside shifts right. Both surprises come from acting on the input.
Factor before reading the shift
sin(2x − 60°) is sin(2(x − 30°)), a shift of 30° rather than 60°.
A negative a
A negative coefficient flips the wave upside down. The amplitude is still its size, never negative.
Four parameters
In y = A sin(B(x − C)) + D, A is amplitude, 2π/B the period, C the horizontal shift, D the midline. Four numbers determine the curve, and each is separately readable from a graph.
Amplitude is half the swing
It measures from the midline to a peak, so it is half the difference between maximum and minimum. The midline is their average, and both are extracted from data the same way.
Factor before reading the shift
sin(2x − π) must be read as sin(2(x − π/2)), a shift of π/2 rather than π. Failing to factor out the B scales the shift wrongly, and it is the standard error.
Graph with five points
Mark the start of a cycle, the quarter points and the end. Those five give the zeros, peak and trough, and joining them produces an accurate sketch far faster than a table.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Rosa Read a Wave
Rosa analyses y = 3 sin(2x − 90°) + 1.
- Step 1
The 3 sits outside the sine, so the amplitude is 3.
Step 4: Your Turn
Practice makes it stick.
The Period
Problem 1 of 2
y = sin(4x) in degrees. What is the period?
degrees
The Shift
Problem 2 of 2
y = sin(2x − 90°). What is the phase shift, in degrees?
degrees
Read the Sinusoid
1 of 8
y = 7 sin x. Amplitude?
2 of 8
y = sin x + 4. Midline value?
3 of 8
y = sin(2x) in degrees. Period?
4 of 8
y = 3 sin x + 5. Maximum value?
5 of 8
y = 2 sin x − 1. Minimum value?
6 of 8
y = −4 sin x. What is the amplitude?
7 of 8
Sort each change by the direction it acts in.
Tap something to move it.
- Empty
- Empty
8 of 8
y = sin(6x) in degrees. Period?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = cos(3x) in degrees. What is the period?
Question 2 of 2
Can an amplitude be negative?
What You Learned
- A sinusoid is a sin(b(x − c)) + d, with one feature per letter.
- The period is 360 ÷ b, and horizontal changes behave backwards.
- Factor the inside before reading the phase shift.