Introduction
The basic sine wave is nice, but rigid. It always goes from -1 to 1, and it always takes to repeat. Real-world waves (sound, light, tides) come in all shapes and sizes.
Prerequisite Connection
You can graph the basic sine and cosine waves and know their key properties: period , amplitude 1, range .
Today's Increment
We learn to stretch the wave vertically (Amplitude A) and horizontally (Period via B) using .
Why This Matters
In Calculus, the Chain Rule tells us how to take the derivative of composite functions like . The "B" pops out to the front: .
Amplitude (Vertical Stretch)
The coefficient A in controls the vertical stretch. The amplitude is always positive: .
Period (Horizontal Stretch/Compression)
The coefficient B inside the function controls the speed.
Frequency is how many cycles happen in .
Period is the length of one cycle.
Notice that makes the wave go twice as fast, so the period is cut in half ().
Worked Examples
Example 1: Analyzing an Equation
Find the amplitude and period of .
Identify Amplitude
Amplitude is the absolute value of A. .
Identify Period
Use the formula with .
.
Example 2: Writing an Equation
Write a sine equation with amplitude 2 and period .
Find B
We know .
Multiply by B: .
Divide by : .
Assemble Equation
or .
Example 3: Graph to Equation (Advanced)
Find the equation of the sine wave shown below.
Find Amplitude (A)
The graph goes up to 3 and down to -3.
.
Find Period and B
The cycle completes at .
Use :.
Write Equation
Common Pitfalls
Negative Amplitude?
Amplitude is a distance, so it is always positive. If , the amplitude is 5. The negative sign just means the graph is reflected (flipped).
Confusing B with Period
is NOT the period. is the frequency (how many cycles fit in ). You must always divide: .
Practice Quiz
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