Function Transformations

Vertical Shifts

• Function: f(x) + 2
  • Shift: Moves the graph up 2 units
• Function: f(x) - 3
  • Shift: Moves the graph down 3 units

Horizontal Shifts

• Function: f(x - 4)
  • Shift: Moves the graph 4 units to the right
• Function: f(x + 3)
  • Shift: Moves the graph 3 units to the left

Reflections

• Function: -f(x)
  • Reflection: Over the x-axis
• Function: f(-x)
  • Reflection: Over the y-axis
• Function: -f(-x)
  • Reflection: Over the origin

Scaling

• Vertical Stretch: 2f(x)
  • Stretches the graph vertically by a factor of 2
• Vertical Shrink: (1/2)f(x)
  • Shrinks the graph vertically by a factor of 2
• Horizontal Shrink: f(2x)
  • Shrinks the graph horizontally by a factor of 2
• Horizontal Stretch: f(1/2 x)
  • Stretches the graph horizontally by a factor of 2

Examples

Example 1: Parabola (f(x) = x²)

• Function: x² + 3
  • Transformation: Vertical shift up 3 units
• Function: x² - 2
  • Transformation: Vertical shift down 2 units

Example 2: Absolute Value (f(x) = |x|)

• Function: |x + 2|
  • Transformation: Horizontal shift left 2 units
• Function: |x - 3|
  • Transformation: Horizontal shift right 3 units

Example 3: Square Root (f(x) = √x)

• Function: -√x
  • Transformation: Reflects over the x-axis
• Function: √-x
  • Transformation: Reflects over the y-axis
• Function: -√-x
  • Transformation: Reflects over the origin

Graphing Using Points

Example: Absolute Value Function

• Functions:
  • y = |x|
  • y = 2|x|
  • y = 1/2|x|
• Key Points: (x, y)
  • |x|: (0,0), (1,1), (2,2), (-1,1), (-2,2)
  • 2|x|: (0,0), (1,2), (2,4), (-1,2), (-2,4)
  • 1/2|x|: (0,0), (1,0.5), (2,1), (-1,0.5), (-2,1)
• Transformation Summary:
  • Vertical Stretch: Multiples y values by factor of 2
  • Vertical Shrink: Halves y values

Quadrant Translations by Sign

• Quadrant 1: (+x, +y)
• Quadrant 2: (-x, +y)
• Quadrant 3: (-x, -y)
• Quadrant 4: (+x, -y)
• Verify direction by setting inside of function to zero

Additional Examples

Example 1: Translate & Graph

• Function: (x - 2)² + 3
  • Transformation: Horizontal shift right 2, vertical shift up 3

Example 2: Translate & graph Negative Parabola

• Function: 3 - (x + 2)²
  • Transformation: Vertical shift up 3, horizontal shift left 2
  • Result: Parabola opens downwards

Example 3: Translate & Reflect Square Root Function

• Function: 4 - √(3 - x)
  • Transformation: Reflects over x-axis and y-axis, horizontal shift right 3
  • Points:
    • (3, 4)
    • (2, 3)
    • (-1, 2)
    • (-6, 1)