Going round the circle repeatedly makes the coordinates rise and fall forever. That is the wave.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Amplitude
In y = a sin(bx), the a stretches the wave vertically. Its height above the midline is |a|.
Period
The b compresses it horizontally. The period is 2π ÷ b — one complete cycle.
Midline
Adding a constant lifts the whole wave. That level is the midline it oscillates about.
Reading a wave from its equation
In y = A sin(B(x − C)) + D, A is the amplitude, the period is 2π/B, C is the horizontal shift and D the vertical shift. Four numbers determine the entire curve.
Amplitude is half the total swing
Amplitude is the distance from the midline to a peak, not from trough to peak. A wave oscillating between 3 and 9 has midline 6 and amplitude 3. Amplitude is always taken positive; a negative A reflects the curve.
The midline is the average
D is the midline — the horizontal line about which the wave oscillates. It is the average of the maximum and minimum, which is how you extract it from data rather than from an equation.
Sine and cosine are the same curve, shifted
cos x = sin(x + π/2). Any sinusoid can be written with either function, so a phase shift can be traded for a change of function. Choosing whichever gives the simpler shift is a legitimate move.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1.68)
Step 3: Watch an Example
One step at a time.
Watch Nadia Read y = 3 sin(2x) + 1
Nadia describes a wave from its equation.
- Step 1
The 3 is the amplitude, so it rises 3 above and falls 3 below the midline.
Step 4: Your Turn
Practice makes it stick.
The Amplitude
Problem 1 of 2
y = 5 sin x. What is the amplitude?
The Maximum
Problem 2 of 2
y = 2 sin x + 3. What is the maximum value?
Read the Wave
1 of 4
y = 4 sin x. Amplitude?
2 of 4
y = sin x + 6. Midline?
3 of 4
y = 3 sin x − 2. Minimum value?
4 of 4
y = sin(4x). Period is 2π divided by what?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 6 sin x + 2. Maximum value?
Question 2 of 2
What does the coefficient of x inside sine control?
What You Learned
- Amplitude is the height above the midline.
- The period is 2π divided by the coefficient of x.
- An added constant sets the midline.