Changes made inside the function act on the input, so they are horizontal. Changes made outside act on the output, so they are vertical.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Period
In y = sin(bx) the period is 360° ÷ b. A bigger b means more cycles in the same space.
Horizontal changes run backwards
Doubling b halves the period, and subtracting c shifts the graph right, not left. Both surprises come from the change acting on the input.
Phase shift
For y = sin(bx − c) the phase shift is c ÷ b. Factor before reading it: sin(2x − 60°) is sin(2(x − 30°)), a shift of 30°, not 60°.
Reading all four
From y = a sin(bx − c) + d: a is amplitude, 360 ÷ b is period, c ÷ b is phase shift, and d is midline.
The inside coefficient squeezes
In sin(Bx), a larger B compresses the wave horizontally, giving period 2π/B. It is counter-intuitive that multiplying by a larger number makes the graph narrower, and it is the same inside-changes-run-backwards behaviour as any transformation.
Phase shift needs the factored form
sin(2x − π) shifts by π/2, not π, because it must be read as sin(2(x − π/2)). Factoring the B out before reading the shift is the step most often skipped, and skipping it gives an answer wrong by a factor of B.
Frequency is the reciprocal of period
Frequency counts cycles per unit, period measures units per cycle. Sound, radio and alternating current are all specified by frequency, so being able to move between the two is routinely necessary.
Graph by marking five points
Mark the start of a cycle, the quarter points and the end. Those five points give the zeros, peak and trough, and joining them smoothly produces an accurate sketch far faster than plotting a table.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Lena Read y = 3 sin(2x − 90°) + 1
Lena needs all four features of this wave.
- Step 1
The 3 sits outside the sine, so it is the amplitude.
Step 4: Your Turn
Practice makes it stick.
The Period
Problem 1 of 2
y = sin(4x). What is the period, in degrees?
degrees
The Tide
Problem 2 of 2
A tide is modelled by y = 2 sin(30t) + 5, with t in hours. How many hours is one full cycle?
hours
Squeeze and Slide
1 of 8
y = sin(2x). Period in degrees?
2 of 8
y = cos(6x). Period in degrees?
3 of 8
y = sin(x − 40°). Phase shift in degrees, right being positive?
4 of 8
y = sin(3x − 60°). Phase shift in degrees?
5 of 8
y = 5 sin(2x) + 3. Maximum value?
6 of 8
A wheel turns once every 20 seconds. What is b if t is in seconds and the model uses degrees?
7 of 8
Sort each change by whether it moves the graph horizontally or vertically.
Tap something to move it.
- Empty
- Empty
8 of 8
y = sin(0.5x). Period in degrees?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = cos(3x). What is the period, in degrees?
Question 2 of 2
In y = sin(2x − 80°), what is the phase shift in degrees?
What You Learned
- Inside the function is horizontal, outside is vertical.
- The period is 360° ÷ b.
- Factor the inside before reading the phase shift: it is c ÷ b.