📌 What are Function Transformations?

Function transformations are changes made to the graph of a parent function. These changes can shift, stretch, compress, or reflect the graph without changing its basic shape.

Complete Form:

\( g(x) = a \cdot f(b(x - h)) + k \)

Vertical Shifts (Up and Down):

\( g(x) = f(x) + k \)

📝 Examples - Vertical Shifts:

• \( g(x) = x^2 + 3 \) → Parabola shifted UP 3 units
• \( g(x) = x^2 - 5 \) → Parabola shifted DOWN 5 units
• \( g(x) = |x| + 2 \) → V-shape shifted UP 2 units
• \( g(x) = \sqrt{x} - 4 \) → Square root shifted DOWN 4 units

Horizontal Shifts (Left and Right):

\( g(x) = f(x - h) \)

📝 Examples - Horizontal Shifts:

• \( g(x) = (x - 4)^2 \) → Parabola shifted RIGHT 4 units
• \( g(x) = (x + 2)^2 \) → Parabola shifted LEFT 2 units
• \( g(x) = |x - 5| \) → V-shape shifted RIGHT 5 units
• \( g(x) = \sqrt{x + 3} \) → Square root shifted LEFT 3 units

📝 Combined Shifts Example:

Describe the transformations: \( g(x) = (x + 3)^2 - 5 \)

Analysis:

• Parent function: \( f(x) = x^2 \)
• \( (x + 3) \) → Horizontal shift LEFT 3 units
• \( -5 \) → Vertical shift DOWN 5 units
• New vertex: \( (-3, -5) \) (was at origin)

Reflection Over the x-axis:

\( g(x) = -f(x) \)

📝 Examples - Reflection over x-axis:

• \( g(x) = -x^2 \) → Parabola opens downward
• \( g(x) = -|x| \) → V-shape opens downward
• \( g(x) = -\sqrt{x} \) → Square root curve below x-axis
• \( g(x) = -2^x \) → Exponential decay instead of growth

Reflection Over the y-axis:

\( g(x) = f(-x) \)

📝 Examples - Reflection over y-axis:

• \( g(x) = (-x)^2 = x^2 \) → Same as original (even function)
• \( g(x) = \sqrt{-x} \) → Square root curve to the left
• \( g(x) = 2^{-x} \) → Exponential reflected over y-axis
• \( g(x) = (-x)^3 = -x^3 \) → Cubic reflected

📝 Both Reflections:

Function: \( g(x) = -f(-x) \)

This reflects over both the x-axis and y-axis (180° rotation about origin)

Example: \( g(x) = -(- x)^3 = -(-x^3) = x^3 \)

Vertical Stretch/Compression:

\( g(x) = a \cdot f(x) \) where \( a \neq 0 \)

📝 Examples - Vertical Dilation:

• \( g(x) = 3x^2 \) → Parabola stretched by 3 (narrower)
• \( g(x) = \frac{1}{2}x^2 \) → Parabola compressed by ½ (wider)
• \( g(x) = 4|x| \) → V-shape stretched by 4
• \( g(x) = -2x^2 \) → Parabola stretched by 2 AND reflected

Horizontal Stretch/Compression:

\( g(x) = f(bx) \) where \( b \neq 0 \)

📝 Examples - Horizontal Dilation:

• \( g(x) = (2x)^2 = 4x^2 \) → Parabola compressed by ½
• \( g(x) = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2 \) → Parabola stretched by 2
• \( g(x) = |3x| \) → V-shape compressed by ⅓
• \( g(x) = \sqrt{4x} = 2\sqrt{x} \) → Square root compressed by ¼

Order of Operations for Transformations:

When multiple transformations are applied, follow this order:

1. Horizontal shifts and stretches/compressions (inside the function)
2. Reflections
3. Vertical stretches/compressions (multiply outside)
4. Vertical shifts (add/subtract outside)

📝 Example 1 - Multiple Transformations:

Describe: \( g(x) = -2(x - 3)^2 + 5 \)

Parent function: \( f(x) = x^2 \)

Transformations in order:

1. \( (x - 3) \) → Horizontal shift RIGHT 3 units
2. Negative sign → Reflection over x-axis
3. \( 2 \) → Vertical stretch by factor of 2
4. \( +5 \) → Vertical shift UP 5 units

Result: Parabola opens downward, vertex at \( (3, 5) \)

📝 Example 2 - Complex Transformation:

Describe: \( g(x) = -\frac{1}{2}|3(x + 4)| - 2 \)

Parent function: \( f(x) = |x| \)

Transformations:

1. \( (x + 4) \) → Horizontal shift LEFT 4 units
2. \( 3 \) inside → Horizontal compression by factor of ⅓
3. Negative sign → Reflection over x-axis
4. \( \frac{1}{2} \) → Vertical compression by factor of ½
5. \( -2 \) → Vertical shift DOWN 2 units

Result: V-shape opens downward, vertex at \( (-4, -2) \)

📝 Example 3 - Writing Transformed Function:

Write the equation for \( f(x) = x^2 \) after:

• Shift right 5 units
• Vertical stretch by factor of 3
• Shift down 7 units

Solution:

Start with: \( f(x) = x^2 \)
Shift right 5: \( (x - 5)^2 \)
Stretch by 3: \( 3(x - 5)^2 \)
Shift down 7: \( 3(x - 5)^2 - 7 \)

Answer: \( g(x) = 3(x - 5)^2 - 7 \)

Step-by-Step Process:

1. Identify the parent function (basic shape)
2. Look inside the function for horizontal changes:
   • Shifts: \( (x - h) \) or \( (x + h) \)
   • Stretches/compressions: coefficient of \( x \)
   • Reflections: negative sign on \( x \)
3. Look outside the function for vertical changes:
   • Stretches/compressions: coefficient multiplying the function
   • Reflections: negative sign in front
   • Shifts: constant added/subtracted
4. Describe all transformations in a logical order

📝 Practice Examples:

1. \( g(x) = \sqrt{x + 6} - 3 \)

Parent: \( f(x) = \sqrt{x} \)
Horizontal shift LEFT 6, Vertical shift DOWN 3

2. \( g(x) = 4(x - 1)^3 \)

Parent: \( f(x) = x^3 \)
Horizontal shift RIGHT 1, Vertical stretch by 4

3. \( g(x) = -|x - 2| + 5 \)

Parent: \( f(x) = |x| \)
Horizontal shift RIGHT 2, Reflection over x-axis, Vertical shift UP 5

4. \( g(x) = \frac{1}{3}(2x)^2 + 1 \)

Parent: \( f(x) = x^2 \)
Horizontal compression by ½, Vertical compression by ⅓, Vertical shift UP 1

⚡ Quick Summary

• Vertical shift: \( f(x) + k \) (up if \( k > 0 \), down if \( k < 0 \))
• Horizontal shift: \( f(x - h) \) (right if \( h > 0 \), left if \( h < 0 \))
• Reflection over x-axis: \( -f(x) \)
• Reflection over y-axis: \( f(-x) \)
• Vertical stretch/compression: \( a \cdot f(x) \) (\( |a| > 1 \) = stretch, \( 0 < |a| < 1 \) = compression)
• Horizontal stretch/compression: \( f(bx) \) (\( |b| > 1 \) = compression, \( 0 < |b| < 1 \) = stretch)
• Inside changes affect \(x\) (horizontal, often opposite)
• Outside changes affect \(y\) (vertical)
• Apply transformations in order: horizontal → reflections → vertical stretch → vertical shift

📚 Transformation Rules Table