Lesson 2.3

Vertical Stretches and Compressions

Multiplying the output to stretch graphs tall or squash them flat. Why is twice as tall.

Introduction

Prerequisite Connection: You know that multiplying by makes numbers bigger, and multiplying by makes them smaller. We are now applying this scalar multiplication to the y-coordinates of a graph.

Today's Increment: We are learning "Non-Rigid Transformations." The shape of the graph actually changes. It get taller (stretch) or shorter (compression).

Why This Matters for Calculus: In physics, represents a wave with 3 times the amplitude. Understanding scaling is vital for modeling forces, sound waves, and economic growth multipliers.

Explanation of Key Concepts

Vertical Scaling

Stretch (

)

The graph gets taller and narrower. Every y-value is multiplied by . Points move away from the x-axis.

Compression (

)

The graph gets shorter and wider. Every y-value is shrunk. Points move closer to the x-axis.

Mapping Rule: (x, y) → (x, a·y)

Worked Examples

Level: Basic

Example 1: The Stretch

Graph relative to .

Analysis

Multiply every output by 2.

Point Check

(The vertex is anchored)

Level: Intermediate

Example 2: The Squish

Graph .

Interpretation

. The graph rises half as fast. It looks "wider" or "flatter."

Level: Advanced (Calculus Prep)

Example 3: Order Matters

Graph .

Order of Operations (PEMDAS)

Multiplication comes before Addition/Subtraction.
First: Stretch vertically by 3.
Second: Shift down by 2.

Point Tracking: (1, 1) → (1, 3) → (1, 1).

Common Pitfalls

Confusing "Wide" with "Horizontal Stretch":
A vertical compression () looks WIDER. Is it a horizontal stretch? For parabolas, yes! But conceptually, focus on the Y-axis change: it was SQUASHED down, which made it spread out like dough.
Applying Stretch to the Shift:
In , the stretch affects the shift because of parentheses! It becomes . Always identify if the shift is inside or outside the multiplication.

Real-World Application

Audio Engineering: Amplitude

Sound waves are modeled by sine functions. The "Vertical Stretch" factor, , is strictly defined as Amplitude (Volume).

When you turn up the volume knob on your stereo, you are literally increasing the value of in the equation . The wave gets taller, moving the speaker cone further, creating louder sound.