🎵 Sine & Cosine Graphing Quiz!

📖 Concepts You Need to Know

🎵 The General Form

f(x) = A · sin(Bx) or f(x) = A · cos(Bx)

A = Amplitude → the "height" of the wave from center. Always positive: |A|.
B = affects Period → how "fast" the wave repeats.

📏 Amplitude

Definition: The amplitude is the maximum distance from the midline (x-axis) to a peak or trough.
Formula: Amplitude = |A|
For f(x) = –3sin(2x), the amplitude = |–3| = 3.
(The negative sign flips the graph upside down, but doesn't change the amplitude!)

⏱️ Period

Definition: The period is the length of one complete cycle of the wave.
Formula: Period = 2π / |B|
For f(x) = –3sin(2x), Period = 2π / 2 = π
This means the graph completes one full wave every π units!

🏔️ Max & Min Points

Maximum value = +|A| (the highest point of the wave)
Minimum value = –|A| (the lowest point of the wave)
For f(x) = –3sin(2x): max = 3, min = –3, so Range = [–3, 3]
Domain of all trig functions = all real numbers = (–∞, ∞)

💡 Worked Example: f(x) = –3sin(2x)

Amplitude: |–3| = 3 (graph goes 3 units up and down)

Period: 2π / 2 = π ≈ 3.14 (one full wave every π units)

Key x-values for one period: 0, π/4, π/2, 3π/4, π

At x=0: –3sin(0) = 0 | At x=π/4: –3sin(π/2) = –3 (MIN) | At x=π/2: –3sin(π) = 0

At x=3π/4: –3sin(3π/2) = 3 (MAX) | At x=π: –3sin(2π) = 0

Domain: (–∞, ∞) Range: [–3, 3]

〰 Sine & Cosine Quiz! 〰

🎵 The General Form

📏 Amplitude

⏱️ Period

🏔️ Max & Min Points

💡 Worked Example: f(x) = –3sin(2x)