Graph Transformations

Graph Transformations: The Shape-Shifting Adventure! 🎢

Imagine you have a bouncy rubber band. You can stretch it, squeeze it, flip it, or do all three at once! That’s exactly what we do with trigonometric graphs. Let’s learn how to become graph magicians!

The Universal Metaphor: The Rubber Band Wave 🌊

Think of a sine wave like a jump rope. When you swing it:

• Pull the ends apart → the wave gets wider (horizontal stretch)
• Pull it up and down → the wave gets taller (vertical stretch)
• Flip it upside down → the wave is reflected

Every transformation is just a different way of playing with your rope!

1. Horizontal Stretching: Making Waves Wider or Narrower

What Is It?

Horizontal stretching changes how fast the wave repeats. It makes the wave wider or skinnier.

The Magic Formula

$y = \sin(Bx)$

The number B controls horizontal stretch:

• B < 1 → Wave stretches WIDER (lazy wave)
• B > 1 → Wave squeezes NARROWER (hyper wave)

Simple Example

Original: y = sin(x) — completes one cycle in 2π

Stretched: y = sin(0.5x) — takes TWICE as long to complete one cycle!

Think of it like this: If you walk at half speed, your journey takes twice as long.

Period = 2π ÷ B

If B = 0.5:
Period = 2π ÷ 0.5 = 4π
(Wave is now TWICE as wide!)

If B = 2:
Period = 2π ÷ 2 = π
(Wave is now HALF as wide!)

Visual Idea

graph TD
    A["sin#40;x#41;"] --> B["Normal wave<br>Period = 2π"]
    C["sin#40;0.5x#41;"] --> D["WIDE wave<br>Period = 4π"]
    E["sin#40;2x#41;"] --> F["NARROW wave<br>Period = π"]

Key Insight 💡

Bigger B = Faster cycles = Narrower wave

It’s backwards from what you’d expect! A bigger number inside makes the wave squish together.

2. Vertical Stretching: Making Waves Taller or Shorter

What Is It?

Vertical stretching changes the height of the wave. The peaks go higher, the valleys go lower!

The Magic Formula

$y = A \cdot \sin(x)$

The number A controls vertical stretch:

• A > 1 → Wave gets TALLER
• A < 1 → Wave gets SHORTER

Simple Example

Original: y = sin(x) — peaks at 1, valleys at -1

Stretched: y = 3sin(x) — peaks at 3, valleys at -3!

Think of it like turning up the volume knob on music. The wave gets LOUDER (bigger amplitude)!

Amplitude = |A|

If A = 3:
Wave reaches from -3 to +3
(THREE times taller!)

If A = 0.5:
Wave reaches from -0.5 to +0.5
(HALF as tall!)

Visual Idea

graph TD
    A["sin#40;x#41;"] --> B["Height: 1"]
    C["2sin#40;x#41;"] --> D["Height: 2<br>DOUBLE!"]
    E["0.5sin#40;x#41;"] --> F["Height: 0.5<br>HALF!"]

Key Insight 💡

The A number = The new height of peaks

If A = 5, the wave reaches up to 5 and down to -5. Simple!

3. Reflections: Flipping the Wave

What Is It?

A reflection flips the wave like a pancake! There are two types:

Vertical Reflection (Flip Upside Down)

$y = -\sin(x)$

The negative sign in front flips the wave upside down.

Original: Starts going UP from zero Reflected: Starts going DOWN from zero

It’s like looking at the wave in a puddle — the reflection is upside down!

Horizontal Reflection (Flip Left-Right)

$y = \sin(-x)$

The negative sign inside flips the wave left to right.

For sine: sin(-x) = -sin(x) — same as vertical flip! For cosine: cos(-x) = cos(x) — no visible change!

Simple Example

y = sin(x) → Starts at 0, goes UP first
y = -sin(x) → Starts at 0, goes DOWN first

The wave is UPSIDE DOWN!

Visual Idea

graph TD
    A["sin#40;x#41;"] --> B["Starts UP ⬆️"]
    C["-sin#40;x#41;"] --> D["Starts DOWN ⬇️<br>FLIPPED!"]

Key Insight 💡

Negative outside = Flip up/down Negative inside = Flip left/right

4. Combined Transformations: Mixing the Magic!

What Is It?

Now we put it ALL together! A wave can be stretched, squeezed, AND flipped at the same time.

The Master Formula

$y = A \cdot \sin(Bx)$

Where:

• A = Vertical stretch (and flip if negative)
• B = Horizontal stretch

Simple Example: y = -2sin(3x)

Let’s break it down step by step:

1. A = -2 → Wave is 2 units tall AND flipped upside down
2. B = 3 → Wave is squeezed to 1/3 width (Period = 2π/3)

Step-by-step breakdown:

Original: sin(x)
↓
Horizontal squeeze: sin(3x)
Period goes from 2π to 2π/3
↓
Vertical stretch: 2sin(3x)
Height goes from 1 to 2
↓
Reflection: -2sin(3x)
Flip it upside down!

Another Example: y = 0.5cos(0.5x)

1. A = 0.5 → Wave is only half as tall
2. B = 0.5 → Wave is twice as wide (Period = 4π)

Visual Flow

graph TD
    A["Start: sin#40;x#41;"] --> B["Apply B<br>Horizontal change"]
    B --> C["Apply A<br>Vertical change"]
    C --> D["Check sign of A<br>Flip if negative"]
    D --> E["Final transformed graph!"]

Key Insight 💡

Order doesn’t matter for the final result! But thinking about each change separately makes it easier.

5. Writing Equations from Graphs: Detective Work!

What Is It?

Now YOU become the detective! Given a picture of a wave, figure out the equation.

The Detective Checklist

1. Find the Amplitude (A):

   • How high does it go? That’s A!
   • If it goes from -3 to +3, A = 3

2. Find the Period:

   • How long until the wave repeats?
   • Then B = 2π ÷ Period

3. Check for Reflection:

   • Does it start by going DOWN? Add a negative!

Simple Example: Mystery Graph

You see a wave that:

• Goes from -2 to +2 (height = 2)
• Repeats every π units
• Starts going DOWN

Detective Work:

Step 1: Find A
Wave goes from -2 to +2
A = 2

Step 2: Find B
Period = π
B = 2π ÷ π = 2

Step 3: Check flip
Starts going DOWN!
Add negative sign.

ANSWER: y = -2sin(2x)

Another Example

You see a cosine wave that:

• Goes from -0.5 to +0.5
• Repeats every 4π units

Detective Work:

Step 1: A = 0.5 (height)
Step 2: B = 2π ÷ 4π = 0.5
Step 3: Starts at a peak (normal cosine)

ANSWER: y = 0.5cos(0.5x)

The Formula Finder

graph TD
    A["Look at wave"] --> B["Measure height<br>Find A"]
    B --> C["Measure one cycle<br>Find Period"]
    C --> D["Calculate B<br>B = 2π ÷ Period"]
    D --> E["Check direction<br>Add - if flipped"]
    E --> F["Write equation!"]

Key Insight 💡

Always ask three questions:

1. How TALL? → A
2. How WIDE? → Period → B
3. Which WAY first? → Sign

Quick Reference Table

You’ve Got This! 🎉

Remember the jump rope! Every transformation is just a different way of playing:

• Pull sideways = Horizontal stretch
• Pull up/down = Vertical stretch
• Flip = Reflection

Now you’re ready to transform ANY trig graph like a pro!

$y = A \cdot \sin(Bx) \text{ or } y = A \cdot \cos(Bx)$

A = height, B = speed, negative = flip!