Lesson 1: Graphing Sine and Cosine — Amplitude and Period
Estimated time: 35-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Sketch the basic graphs of y = sin x and y = cos x over one and two periods
- Identify the key features: domain, range, period, amplitude, intercepts, and max/min
- Determine amplitude |A| and period 2π/|B| from y = A sin(Bx) or y = A cos(Bx)
- Graph transformed sine and cosine functions by hand
The Basic Sine Graph
The graph of y = sin x is a smooth wave that oscillates between −1 and 1.
Key properties of y = sin x:
Domain: all real numbers | Range: [−1, 1] | Period: 2π | Amplitude: 1
Key points per period: (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0)
The sine function starts at 0, rises to 1 at a quarter period, returns to 0 at the half period, drops to −1 at three quarters, and returns to 0 to complete the cycle.
The Basic Cosine Graph
The cosine graph has the same shape as sine, but it starts at its maximum value.
Key properties of y = cos x:
Domain: all real numbers | Range: [−1, 1] | Period: 2π | Amplitude: 1
Key points per period: (0, 1), (π/2, 0), (π, −1), (3π/2, 0), (2π, 1)
In fact, cos x = sin(x + π/2), so cosine is just sine shifted left by π/2.
Amplitude
Amplitude of y = A sin(Bx) or y = A cos(Bx) is |A|. It measures the height from the midline to the peak (or trough).
If A > 1, the graph is vertically stretched. If 0 < A < 1, it is compressed. If A < 0, the graph is reflected across the x-axis.
Example 1: Effect of Amplitude
Compare y = sin x, y = 3 sin x, and y = (1/2) sin x.
y = sin x: oscillates between −1 and 1. Amplitude = 1.
y = 3 sin x: oscillates between −3 and 3. Amplitude = 3.
y = (1/2) sin x: oscillates between −1/2 and 1/2. Amplitude = 1/2.
Example 2: Negative Amplitude (Reflection)
Graph y = −2 cos x.
Solution: Amplitude = |−2| = 2. The negative sign reflects the graph across the x-axis. So it starts at −2 (instead of 2), rises to 0, reaches +2, returns to 0, and back to −2.
Period
Period of y = A sin(Bx) or y = A cos(Bx) is 2π/|B|. It is the horizontal length of one complete cycle.
If B > 1, the period decreases (graph is compressed horizontally). If 0 < B < 1, the period increases (graph is stretched).
Example 3: Finding the Period
Find the period of y = sin(2x).
Solution: B = 2. Period = 2π/2 = π. The graph completes one full cycle in π units.
Example 4: Amplitude and Period Together
For y = 4 cos(3x), identify the amplitude and period, and list the five key points.
Amplitude: |4| = 4 Period: 2π/3
Key points: Divide the period into 4 equal parts: 0, π/6, π/3, π/2, 2π/3
(0, 4), (π/6, 0), (π/3, −4), (π/2, 0), (2π/3, 4)
Graphing Strategy
- Identify A (amplitude) and B (used to find period = 2π/|B|)
- Divide one period into four equal intervals to find five key x-values
- Plot the five key points using the pattern of sin or cos
- Connect with a smooth curve
- Extend the pattern left and right as needed
Check Your Understanding
1. Find the amplitude and period of y = 5 sin(4x).
2. Find the amplitude and period of y = −(1/3) cos(πx).
3. What are the five key points for y = 2 sin(x) over one period [0, 2π]?
Key Takeaways
- Amplitude = |A|: the height from midline to peak.
- Period = 2π/|B|: the length of one complete cycle.
- Sine starts at the midline; cosine starts at its maximum.
- Divide each period into 4 parts to find 5 key points for graphing.
- A negative A value reflects the graph across the x-axis.