Absolute Value Functions - Transformations and Effects
1️⃣ Understand the two cases of absolute value in functions
2️⃣ Learn how
affects the graph (Case 1: Reflect below x-axis)
3️⃣ Learn how
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
The absolute value effect on functions creates two distinct cases with completely different transformations:
Two Cases of Absolute Value
Case 1:
Absolute value on the entire function
Effect: Reflect parts below x-axis upward
Case 2:
Absolute value on x only
Effect: Duplicate right side to left side
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.
{{AbsoluteValueCase1Simulator}}
Case 1:
Key word: REFLECT (not duplicate!)
Case 1 Process:
- Look at the original function
- 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
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.
{{AbsoluteValueCase2Simulator}}
Case 2:
Key word: DUPLICATE (not reflect!)
Case 2 Process:
- Look at the original function
- 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)
Critical Differences
Aspect
Location of | |
Action
Focus Area
Result
Case 1:
Around entire function
REFLECT
Below x-axis
No negative y-values
Case 2:
Around x only
DUPLICATE
Right of y-axis
Even function (symmetric)
Let's see how both cases affect three different types of functions:
{{FunctionExamplesSimulator}}
Function Examples
Example 1: Quadratic Function
Case 1:
The parabola dips below x-axis between and
→ Reflect this dip upward to create a "W" shape
Case 2:
Right side of parabola (x ≥ 0) gets copied to left side
→ Duplicate creates same parabola (already even!)
Example 2: Cubic Function
Case 1:
Cubic goes negative between and
→ Reflect this portion upward
Case 2:
Right side of cubic gets copied to left side
→ Duplicate creates even function
Example 3: Sine Function
Case 1:
Sine waves below x-axis (negative portions)
→ Reflect all negative waves upward
Case 2: for
Right side sine wave gets copied to left
→ Duplicate creates even sine function
🧠 Memory Aids
Case 1:
"Function in jail" 🔒
Cannot go below zero
REFLECT negative parts
Case 2:
"X in jail" 🔐
X cannot be negative
DUPLICATE right to left
🎯 Key Takeaways
- Case 1 (): Absolute value around entire function → REFLECT parts below x-axis
- Case 2 (): 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