Lesson 1.2

Vertical Transformations

What happens when we multiply by a number? We stretch it, shrink it, or flip it upside down.

Introduction

The parent function is just the beginning. By multiplying our function by a constant, we can stretch it to reach new heights or compress it to fit wider spaces, modeling everything from steep arches to shallow valleys.

Past Knowledge

You know the parent function has a vertex at (0,0) and opens up.

Today's Goal

We explore functions of the form . We will learn how the value of changes the shape of the graph.

Future Success

This is the first step to mastering Vertex Form , which combines all transformations.

Key Concepts

The Role of 'a' in

The coefficient controls the width and direction of the parabola.

Vertical Stretch
If is a number like 2, 3, or 10, the y-values get multiplied, making the graph grow faster. The parabola looks narrower.
Vertical Compression
If is a fraction like 1/2 or 0.1, the y-values grow slower. The parabola looks wider.
Reflection
If is negative, the graph flips over the x-axis and opens downward.

Parent function (dashed gray) vs. Transformations.

Worked Examples

Example 1: Vertical Stretch

Basic

Graph . Describe the transformation.

Compare to parent function

Since and , this is a Vertical Stretch by a factor of 2.

Create a Table

x	x²	2x²
-1	1	2
0	0	0
1	1	2
2	4	8

Example 2: Vertical Compression

Concept

Graph . Describe the transformation.

Identify 'a'

. Since , this is a Vertical Compression (Shrink) by a factor of 1/2.

Example 3: Combined Reflection & Stretch

Advanced

Graph .

Identify Transformations

Negative sign (): Reflects across x-axis (opens down).
Number 3 (): Vertical Stretch by 3 using (narrow).

Key Points

Vertex:

x=1:

x=2:

Common Pitfalls

"Bigger Number = Wider Graph"

Students often think is "bigger" so it should be wider. It's the opposite! A large 'a' value shoots the graph up faster, making it narrower.

Negative Signs Inside the Square

is a vertical reflection. simplifies to (no change). Be careful where the negative sign is!

Real-Life Applications

Architecture: St. Louis's Gateway Arch is a catenary curve (very similar to a parabola). When architects design arches, they adjust the "a" value to make the arch taller (stretch) or wider (compression) to span different distances or reach specific heights.