Absolute Value Functions - Transformations and Effects

1️⃣ Understand the two cases of absolute value in functions

2️⃣ Learn how $|f(x)|$ affects the graph (Case 1: Reflect below x-axis)

3️⃣ Learn how $f(|x|)$ affects the graph (Case 2: Duplicate right side to left)

4️⃣ Apply transformations to various function types

5️⃣ Master the difference between reflection and duplication

🔍 Overview: Two Cases of Absolute Value

The absolute value effect on functions creates two distinct cases with completely different transformations:

Two Cases of Absolute Value

Case 1: $|f(x)|$

Absolute value on the entire function

Effect: Reflect parts below x-axis upward

Case 2: $f(|x|)$

Absolute value on x only

Effect: Duplicate right side to left side

1️⃣ Case 1: $|f(x)|$ - Reflect Below X-axis

When the absolute value is applied to the entire function, we take any part of the curve that is below the x-axis and reflect it upward.

Case 1: $y = |f(x)|$

Original: $y = f(x)$ → Transformed: $y = |f(x)|$

Key word: REFLECT (not duplicate!)

Case 1 Process:

Look at the original function $f(x)$
Identify all parts that are below the x-axis (negative y-values)
Reflect these parts upward - flip them across the x-axis
Keep all parts above the x-axis exactly the same
Result: The new graph has no negative y-values

2️⃣ Case 2: $f(|x|)$ - Duplicate Right Side to Left

When the absolute value is applied to x only, we take the entire curve on the right side of the y-axis and duplicate it (copy it) to the left side.

Case 2: $y = f(|x|)$

Original: $y = f(x)$ → Transformed: $y = f(|x|)$

Key word: DUPLICATE (not reflect!)

Case 2 Process:

Look at the original function $f(x)$
Focus on the part that is to the right of the y-axis (positive x-values)
Copy this entire right side - whether it's above or below the x-axis
Duplicate it to the left side of the y-axis
Result: The graph becomes symmetric about the y-axis (even function)

📊 Comparison: Key Differences Between Cases

Critical Differences

Aspect

Location of | |

Action

Focus Area

Result

Case 1: $|f(x)|$

Around entire function

REFLECT

Below x-axis

No negative y-values

Case 2: $f(|x|)$

Around x only

DUPLICATE

Right of y-axis

Even function (symmetric)

🔄 Examples: Three Functions Transformed

Let's see how both cases affect three different types of functions:

Function Examples

Example 1: Quadratic Function $f(x) = x^2 - 4$

Case 1: $|f(x)| = |x^2 - 4|$

The parabola dips below x-axis between $x = -2$ and $x = 2$

→ Reflect this dip upward to create a "W" shape

Case 2: $f(|x|) = |x|^2 - 4 = x^2 - 4$

Right side of parabola (x ≥ 0) gets copied to left side

→ Duplicate creates same parabola (already even!)

Example 2: Cubic Function $f(x) = x^3 - x$

Case 1: $|f(x)| = |x^3 - x|$

Cubic goes negative between $x = 0$ and $x = 1$

→ Reflect this portion upward

Case 2: $f(|x|) = |x|^3 - |x|$

Right side of cubic gets copied to left side

→ Duplicate creates even function

Example 3: Sine Function $f(x) = \sin(x)$

Case 1: $|f(x)| = |\sin(x)|$

Sine waves below x-axis (negative portions)

→ Reflect all negative waves upward

Case 2: $f(|x|) = \sin(|x|) = \sin(x)$ for $x ≥ 0$

Right side sine wave gets copied to left

→ Duplicate creates even sine function

🧠 Memory Aids

Case 1: $|f(x)|$

"Function in jail" 🔒

Cannot go below zero

REFLECT negative parts

Case 2: $f(|x|)$

"X in jail" 🔐

X cannot be negative

DUPLICATE right to left

🎯 Key Takeaways

Case 1 ($|f(x)|$): Absolute value around entire function → REFLECT parts below x-axis
Case 2 ($f(|x|)$): Absolute value around x only → DUPLICATE right side to left
Remember: "REFLECT vs DUPLICATE" - completely different transformations!
Case 1 result: No negative y-values (output always ≥ 0)
Case 2 result: Even function (symmetric about y-axis)
The position of the absolute value symbols determines which case applies

Transformations of Functions

Applying Absolute Value on Functions