Horizontal Stretches and Compressions
The "Inside is Opposite" rule strikes again. Why multiplying by 3 actually shrinks the graph horizontally.
Introduction
Prerequisite Connection: Remember how moved the graph to the RIGHT +5? Inside changes were opposite. That same logic applies to multiplication.
Today's Increment: We are learning how to handle coefficients inside the function, like . This causes horizontal scaling, but in reverse of what you might expect.
Why This Matters for Calculus: In integration by substitution (u-sub), if you have , you end up dividing by 3 ("multiplying by 1/3"). This lesson is the geometric reason why that happens.
Explanation of Key Concepts
Horizontal Scaling
If , the graph happens "twice as fast." It shrinks horizontally by .
If , the graph is "slowed down." It stretches horizontally by .
Worked Examples
Example 1: The "Fast Forward"
Graph relative to .
- (Same height, half the width)
Example 2: The "Slow Motion"
Graph .
Example 3: The Danger Zone
Graph .
This is the #1 error on Pre-Calculus exams. You see "-6" and assume Right 6. But the "2" is messing everything up.
- b = 2 (Compression by 1/2)
- h = 3 (Shift Right 3)
Common Pitfalls
- The "Right 6" Trap:
As seen in Example 3, in , the shift is only 3. Why? Because you compress the x-axis FIRST, which also compresses the shift distance.
- Mixing up Stretches:
is Vertical Stretch. is Horizontal Compression. One makes it tall, the other makes it skinny.
Real-World Application
Physics: Period and Frequency
In simple harmonic motion (springs, pendulums), the function is .
The coefficient is a Horizontal Compression factor. In physics, we call this angular frequency.
- High (big compression) = Fast oscillation (High Pitch).
- Low (big stretch) = Slow oscillation (Low Pitch).
Practice Quiz
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