Lesson 2.3

Vertical Stretches and Compressions

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

Introduction

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

Past Knowledge

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 Goal

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

Future Success

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

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-Life Applications

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.