Cogito
AP Precalculus · Chapter 6 · Lesson 2
Sinusoidal Functions
Amplitude, period, midline and phase.
12 problems · about 22 minutes · AP Precalculus 3.5, 3.6, 3.7
What this lesson teaches
The student determines the four features of a sinusoidal function and models periodic data.
- A sinusoid is a sin(b(x − c)) + d, with each letter controlling one feature.
- Amplitude is half the peak-to-trough gap and the midline is their average.
- The period is 2π ÷ b, and the phase shift needs the inside factored first.
Warm Up
Straightforward practice. Get the method working first.
5 problemsMax 30 and min 10. What is the midline value?
Answer 20
Why 20.
Why factor the inside before reading a phase shift?
Answer The shift is c after factoring, not the constant as written.
Why Factoring reveals the true shift.
y = 7 sin x. What is the amplitude?
Answer 7
Why The coefficient.
y = sin x + 4. What is the midline value?
Answer 4
Why The added constant.
y = sin(2x) in degrees. What is the period?
Answer 180
Why 360 ÷ 2.
Build It Up
The same ideas with more to keep track of.
3 problemsy = sin(4x) in degrees. What is the period?
Answer 90
Why 360 ÷ 4.
y = 3 sin x + 5. What is the maximum value?
Answer 8
Why Midline plus amplitude.
Max 20 and min 4. What is the amplitude?
Answer 8
Why (20 − 4) ÷ 2.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each parameter with the feature it controls.
Answer a → Amplitude; b → Period; d → Midline
Why Each letter controls exactly one feature.
y = 2 sin x − 1. What is the minimum value?
Answer -3
Why Midline minus amplitude.
The Amplitude: Max 42 and min 2. What is the amplitude?
Answer 20
Why 20.
The Midline: Same values. What is the midline?
Answer 22
Why 22.