Lecture Notes: Graphing Radical Functions

Introduction

• Two types of radical functions: even and odd.
• Each type follows a general pattern.

Even Radical Functions

• Example: Fourth root of x.
• Graph Characteristics:
  • Starts at the origin: (0, 0).
  • Graph shape: upward and to the right.
  • Center can move due to transformations (up, down, left, right).
• Domain:
  • Only includes x-values to the right of the origin (x ≥ 0).
• Range:
  • Only y-values above the x-axis (y ≥ 0).

Odd Radical Functions

• Graph Characteristics:
  • Typically takes an "S" shape.
  • Passes through the origin.
  • X-values can extend infinitely in both directions.
• Domain and Range:
  • Both are all real numbers.

Transformations of Radical Functions

Example 1: Even Radical Function with Transformations

• Original Function: Parent looks like basic even graph.
• Transformation Example:
  • Function: x + 2 moves the graph left 2 units, and -1 moves it down 1 unit.
  • New vertex: Left 2 units, down 1 unit from origin.
• Domain:
  • x ≥ -2.
• Range:
  • y ≥ -1.

Example 2: Odd Radical Function with Transformations

• Transformation Example:
  • Vertical stretch by a factor of 2.
  • Horizontal shift: Right 7 units.
  • Vertical shift: Up 2 units.
• Graph: Still has the same shape, moved from the origin.
• Domain and Range:
  • Both are all real numbers.

Example 3: Even Radical Function with Reflection

• Function: y = -√(x - 4) + 5.
• Transformations:
  • Negative sign reflects over the x-axis.
  • Shift right 4 units and up 5 units.
• Graph Shape: Inverted due to reflection.
• Vertex: Positioned at (4, 5).
• Domain:
  • x ≥ 4.
• Range:
  • y ≤ 5 (because of downward reflection).

Conclusion

• Understanding transformations and their effects on the graph is crucial.
• Differences in behavior between even and odd radical functions.
• Even functions have restricted domains and ranges, while odd functions encompass all real numbers.