Cogito
Trigonometry · Chapter 3 · Lesson 2
Period and Phase Shift
The coefficient inside squeezes; the number inside slides.
12 problems · about 22 minutes · F-TF.B.5, F-IF.C.7
What this lesson teaches
The student determines period and phase shift from an equation of the form y = a sin(bx − c) + d.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = cos(3x). What is the period, in degrees?
Answer 120
Why 120°.
In y = sin(2x − 80°), what is the phase shift in degrees?
Answer 40
Why 40° to the right.
y = sin(2x). Period in degrees?
Answer 180
Why 360 ÷ 2.
y = cos(6x). Period in degrees?
Answer 60
Why 360 ÷ 6.
y = sin(x − 40°). Phase shift in degrees, right being positive?
Answer 40
Why Subtracting inside shifts right.
Build It Up
The same ideas with more to keep track of.
3 problemsy = sin(3x − 60°). Phase shift in degrees?
Answer 20
Why Factor first: 3(x − 20°).
y = 5 sin(2x) + 3. Maximum value?
Answer 8
Why Midline plus amplitude.
A wheel turns once every 20 seconds. What is b if t is in seconds and the model uses degrees?
Answer 18
Why 360 ÷ 20.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each change by whether it moves the graph horizontally or vertically.
Answer Horizontal: The b in sin(bx), The c in sin(x − c) · Vertical: The a in a sin x, The d in sin x + d
Why Inside the function is horizontal; outside is vertical.
y = sin(0.5x). Period in degrees?
Answer 720
Why 360 ÷ 0.5.
The Period: y = sin(4x). What is the period, in degrees?
Answer 90 degrees
Why 90°.
The Tide: A tide is modelled by y = 2 sin(30t) + 5, with t in hours. How many hours is one full cycle?
Answer 12 hours
Why 12 hours.