Cogito
Integrated Math 3 · Chapter 5 · Lesson 3
Modelling Periodic Phenomena
Fitting a wave to something that repeats.
12 problems · about 21 minutes · F-TF.B.5, F-IF.B.4
What this lesson teaches
The student builds sinusoidal models from described periodic data and predicts values.
- Midline is the average of max and min; amplitude is half their gap.
- b is 360 divided by the period.
- Choosing the right starting curve removes the need for a phase shift.
Warm Up
Straightforward practice. Get the method working first.
5 problemsMax 30 and min 10. What is the amplitude?
Answer 10
Why 10.
Data starts at its minimum. What model avoids a phase shift?
Answer A negative cosine, which starts at its lowest point.
Why An upside-down cosine.
Max 20, min 4. Amplitude?
Answer 8
Why (20 − 4) ÷ 2.
Max 20, min 4. Midline?
Answer 12
Why (20 + 4) ÷ 2.
A cycle takes 12 hours. What is b, in degrees per hour?
Answer 30
Why 360 ÷ 12.
Build It Up
The same ideas with more to keep track of.
3 problemsh = 5 sin(60t) + 9. Maximum value?
Answer 14
Why Midline plus amplitude.
h = 5 sin(60t) + 9. Period in units of t?
Answer 6
Why 360 ÷ 60.
h = −20 cos(45t) + 22 at t = 0. What is h?
Answer 2
Why −20 + 22.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the modelling steps in order.
Answer 1. Find the midline from the average of max and min. 2. Find the amplitude from half the gap. 3. Find b from 360 divided by the period. 4. Choose sine or cosine from where the data starts.
Why The starting point matters only once the shape is fixed.
A wheel runs between 1 m and 31 m. Midline in metres?
Answer 16
Why (31 + 1) ÷ 2.
The Amplitude: Max 42 and min 2. What is the amplitude?
Answer 20
Why 20.
The Value of b: A cycle takes 8 minutes. What is b, in degrees per minute?
Answer 45
Why 45.