Introduction
We can stretch the wave vertically to make it louder, or squish it horizontally to make it faster.
Past Knowledge
You can graph the basic sine and cosine waves and know their key properties: period , amplitude 1, range .
Today's Goal
We learn to stretch the wave vertically (Amplitude A) and horizontally (Period via B) using .
Future Success
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
Practice Quiz
Loading...