Complete Lesson: Function Transformation Order of Operations
1. General Rule: Inside vs. Outside Transformations
Transformations are split into two groups:
- Inside the function (horizontal transformations): Affect \( x \)-values
- Outside the function (vertical transformations): Affect \( y \)-values
2. Order of Operations
Horizontal Transformations (applied first, in reverse order)
- Stretch/compression by \( \frac{1}{|b|} \)
\( f(2x) \) compresses horizontally by \( \frac{1}{2} \)
- Reflection if \( b < 0 \)
\( f(-x) \) reflects over \( y \)-axis
- Horizontal shift by \( h \)
\( f(x - 3) \) shifts right 3 units
Vertical Transformations (applied last, in normal order)
- Stretch/compression by \( |a| \)
\( 2f(x) \) stretches vertically by 2
- Reflection if \( a < 0 \)
\( -f(x) \) reflects over \( x \)-axis
- Vertical shift by \( k \)
\( f(x) + 5 \) shifts up 5 units
3. Why Order Matters
Horizontal transformations are reversed:
For \( f(2(x - 3)) \):
\[ \begin{align*} 1. &\ x \rightarrow x - 3 \quad \text{(shift right 3)} \\ 2. &\ x \rightarrow 2(x - 3) \quad \text{(compress by } \frac{1}{2}\text{)} \end{align*} \]Vertical transformations follow natural order:
For \( -2f(x) + 5 \):
\[ \begin{align*} 1. &\ y \rightarrow 2f(x) \quad \text{(stretch by 2)} \\ 2. &\ y \rightarrow -2f(x) \quad \text{(reflect over x-axis)} \\ 3. &\ y \rightarrow -2f(x) + 5 \quad \text{(shift up 5)} \end{align*} \]4. Complete Example: Applying All Transformations
Original function: \( f(x) = \sqrt{x} \)
Transformed function: \( y = -3\sqrt{-2(x + 1)} + 4 \)
Step 1: Horizontal Transformations
\[ \begin{align*} \sqrt{x} & \rightarrow \sqrt{2x} \quad \text{(compress by } \frac{1}{2}\text{)} \\ & \rightarrow \sqrt{-2x} \quad \text{(reflect over y-axis)} \\ & \rightarrow \sqrt{-2(x + 1)} \quad \text{(shift left 1)} \end{align*} \] <--!more-->Step 2: Vertical Transformations
\[ \begin{align*} \sqrt{-2(x + 1)} & \rightarrow 3\sqrt{-2(x + 1)} \quad \text{(stretch by 3)} \\ & \rightarrow -3\sqrt{-2(x + 1)} \quad \text{(reflect over x-axis)} \\ & \rightarrow -3\sqrt{-2(x + 1)} + 4 \quad \text{(shift up 4)} \end{align*} \]5. Key Takeaways
- ✅ Always apply all transformations
- ✅ Horizontal order:
SRS(Stretch → Reflect → Shift) - ✅ Vertical order:
SRS(Stretch → Reflect → Shift) - ✅ Practice with \( f(x) = x \) first
6. Common Pitfalls to Avoid
- ⚠️ Applying shifts before stretches
- ⚠️ Mixing horizontal/vertical order
- ⚠️ Ignoring negative signs (reflections)
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