🔄 Function Transformations
Discover how functions shift, stretch, reflect, and transform on the coordinate plane! Master the art of manipulating function graphs.
📊 Understanding Transformations
A transformation changes the position, size, or orientation of a function's graph. Think of it like moving, stretching, or flipping an image!
The Four Main Types
- Translations: Slide the graph up, down, left, or right
- Reflections: Flip the graph over an axis
- Stretches: Make the graph taller or shorter
- Compressions: Squeeze the graph vertically or horizontally
Parent Function: The simplest function in a family (like $f(x) = x^2$ for parabolas)
We transform the parent function to create new related functions!
⬆️⬇️ Vertical Translations
Adding or subtracting a constant outside the function shifts it vertically!
The Rule
$g(x) = f(x) + k$
- If $k > 0$: shift UP by $k$ units
- If $k < 0$: shift DOWN by $|k|$ units
Example 1: Starting with $f(x) = x^2$
$g(x) = x^2 + 3$ shifts the parabola UP 3 units
The vertex moves from $(0, 0)$ to $(0, 3)$
Example 2
$g(x) = x^2 - 5$ shifts the parabola DOWN 5 units
The vertex moves from $(0, 0)$ to $(0, -5)$
Key Point: Every point on the graph moves the same vertical distance!
➡️⬅️ Horizontal Translations
Changing the input inside the function shifts it horizontally!
The Rule (Watch the Signs!)
$g(x) = f(x - h)$
- If $h > 0$: shift RIGHT by $h$ units
- If $h < 0$: shift LEFT by $|h|$ units
⚠️ Common Mistake!
The sign is opposite what you might expect!
$g(x) = (x - 2)^2$ shifts RIGHT 2 (not left!)
$g(x) = (x + 3)^2$ shifts LEFT 3 (not right!)
Example: $f(x) = |x|$
$g(x) = |x - 4|$ shifts the V-shape RIGHT 4 units
The vertex moves from $(0, 0)$ to $(4, 0)$
🪞 Reflections
Flip the graph over an axis like looking in a mirror!
Reflection over the x-axis
$g(x) = -f(x)$ (negative outside)
Every $y$-value becomes its opposite!
Example: $g(x) = -x^2$ flips the parabola upside down
Reflection over the y-axis
$g(x) = f(-x)$ (negative inside)
The graph flips from left to right!
Example: If $f(x) = 2^x$, then $g(x) = 2^{-x}$ reflects it over the y-axis
Visual Tip
✨ Think of the x-axis as a mirror on the floor (flip up/down)
✨ Think of the y-axis as a mirror on a wall (flip left/right)
📏 Vertical Stretches and Compressions
Multiply the function by a constant to change its vertical size!
The Rule
$g(x) = a \cdot f(x)$ where $a > 0$
- If $a > 1$: vertical stretch (taller, steeper)
- If $0 < a < 1$: vertical compression (shorter, flatter)
Example 1: Stretch
$g(x) = 3x^2$ stretches $f(x) = x^2$ by factor of 3
Point $(2, 4)$ becomes $(2, 12)$ - three times as tall!
Example 2: Compression
$g(x) = \frac{1}{2}|x|$ compresses the absolute value graph
Makes it flatter, wider-looking
Remember: This affects $y$-values only!
🎯 Combining Multiple Transformations
Real-world problems often combine several transformations!
Order of Operations Matters!
Apply transformations in this order:
- Horizontal shifts (inside the function)
- Reflections
- Stretches/compressions
- Vertical shifts (outside the function)
Example: $g(x) = -2(x - 3)^2 + 1$
Starting from $f(x) = x^2$:
- Shift RIGHT 3: $(x - 3)^2$
- Stretch by 2: $2(x - 3)^2$
- Reflect over x-axis: $-2(x - 3)^2$
- Shift UP 1: $-2(x - 3)^2 + 1$
The vertex is at $(3, 1)$ and the parabola opens downward! 🎯
🌟 Real-World Applications
- 📈 Economics: Shift supply/demand curves to model market changes
- 🎨 Computer Graphics: Transform images, rotate objects in games
- 🔬 Physics: Model motion with shifted and scaled functions
- 📊 Data Science: Normalize and scale data for analysis
- 🎵 Music: Audio engineering uses function transformations for effects
🎯 Practice Questions
Master these transformations!
🔥 Challenge Questions
Advanced transformations!