Graph of y = A sin(Bθ+C)+D: amplitude |A|, period 2π/|B|, phase shift −C/B, and vertical translation D。
Example ①
State the amplitude and period of y = 2sin(3θ).
Solution:Amplitude 2, period 2π/3
State the amplitude and period of y = 2sin(3θ).
Solution:Amplitude 2, period 2π/3
Example ②
Find the maximum and minimum values of y = 3cos(θ/2)+1.
Solution:Maximum 4, minimum −2
Find the maximum and minimum values of y = 3cos(θ/2)+1.
Solution:Maximum 4, minimum −2
Key Points
- Period = 2π/|B|, maximum value = |A|+D, minimum value = −|A|+D
Practice Problems
- State the amplitude, period, maximum value, and minimum value of y = −2cos(2θ).
- Find the maximum value of y = sin θ+cos θ.
- Find the maximum value of y = 2sin(θ−π/4)−1 and the value of θ when it occurs (0≤θ≤2π).
Solutions and Explanations
- Answer:Amplitude 2, period π, maximum value 2, minimum value −2
- Answer:√2（expressed as √2 sin(θ+π/4), with amplitude √2）
- Answer:Maximum value 1, θ = 3π/4