Introduction
We now complete our transformation toolkit. We can move the graph up and down (Vertical Translations) or left and right (Phase Shifts). Combining all four transformations gives us the General Sinusoidal Form:
- A Amplitude (Stretch)
- B Frequency (Period)
- C Phase Shift (L/R)
- D Midline (Up/Down)
Prerequisite Connection
You can stretch waves using Amplitude (A) and Period (B) from Lesson 10.2. Now we add translations.
Today's Increment
We add Phase Shift (C) for horizontal movement and Midline (D) for vertical movement.
Why This Matters
In Calculus, you'll use Fourier Series to represent ANY periodic function as a sum of shifted sine waves—the basis for signal processing and music synthesis.
Vertical Shift (The Midline)
The number added at the end, D, moves the graph up or down. The line becomes the new "center" of the wave, called the Midline.
Phase Shift (Horizontal Translation)
The value C inside the parentheses moves the graph left or right.
Be Careful: It is .
- moves Right by .
- moves Left by .
Worked Examples
Example 1: Identify Shifts
Identify the phase shift and vertical shift of .
Vertical Shift (D)
The constant at the end is -4. The graph moves Down 4.
Phase Shift (C)
Inside is . Since the formula is , we interpret plus as left. The graph moves Left .
Example 2: The Factoring Trap
Find the phase shift of .
Factor out B
Before identifying C, we must factor out the coefficient of x.
Identify C
Now we see the shift is actually Right .
Example 3: Write an Equation (Advanced)
Write a cosine equation with amplitude 2, period , phase shift right, and midline .
Find B from Period
.
Assemble Parameters
, , (right), .
Write Equation
Real-World Application
Tides and Temperature
Daily temperatures follow a sine wave. But the average temperature isn't 0 degrees! If the average temp is 70°F, that's a Vertical Shift of +70.
The hottest part of the day isn't at midnight (t=0). It's usually around 3 PM. That delay is a Phase Shift!
Practice Quiz
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