Module 4: Practice Problems
Instructions: Identify the key features and graph each function.
1
Find the amplitude and period of y = 3 sin(2x).
Solution
Amplitude = 3, Period = 2π/2 = π.
2
For y = −2 cos(x − π/3) + 1, find all four parameters.
Solution
Amp = 2, Period = 2π, Phase shift = π/3 right, Vertical shift = 1. Range: [−1, 3].
3
Find the period of y = cos(4x).
Solution
Period = 2π/4 = π/2.
4
List the five key points for y = sin(x) over [0, 2π].
Solution
(0,0), (π/2,1), (π,0), (3π/2,−1), (2π,0).
5
What is the range of y = 4 sin(x) − 2?
Solution
[−2 − 4, −2 + 4] = [−6, 2].
6
Find the period and asymptotes of y = tan(2x).
Solution
Period = π/2. Asymptotes: 2x = ±π/2 + nπ, so x = ±π/4 + nπ/2.
7
Where are the vertical asymptotes of y = csc(x)?
Solution
At x = nπ (where sin x = 0).
8
Find the phase shift of y = 2 sin(3x − π/2).
Solution
C/B = (π/2)/3 = π/6 right.
9
What is the range of y = 2 sec(x)?
Solution
(−∞, −2] ∪ [2, ∞).
10
Find the period of y = cot(x/3).
Solution
Period = π/(1/3) = 3π.