Notes on Graphing Radical Equations

Introduction to Radical Equations

• Focus: Graphing radical equations
• Starting with the parent function:
  [ y = \sqrt{x} ]
• Characteristics of the graph:
  • Starts at the origin (0,0)
  • Increases at a decreasing rate

Domain and Range of y = √x

• Domain:
  • Lowest x-value: 0
  • Highest x-value: ∞
  • Domain: [0, ∞)
• Range:
  • Lowest y-value: 0
  • Highest y-value: ∞
  • Range: 0, ∞)

Transformations of the Parent Function

Negative Square Roots

• y = -√x: Reflects over the x-axis
• y = √-x: Reflects over the y-axis
• y = -√-x: Reflects over the origin

Quadrant Analysis for Transformations

• Helps determine direction of graphs:
  • Positive x & y: Quadrant I
  • Positive x & Negative y: Quadrant IV
  • Negative x & Positive y: Quadrant II
  • Negative x & Negative y: Quadrant III

Vertical and Horizontal Shifts

• Vertical Shifts:
  • Adding to the function shifts it up
  • Subtracting from the function shifts it down
  • Example:
    • y = √x + 2: Starts 2 units above x-axis
    • y = √x - 1: Shifts down 1 unit
• Horizontal Shifts:
  • Inside the radical affects x values
  • Example:
    • y = √(x - 2): Shifts right 2 units
    • y = √(x + 3): Shifts left 3 units

Graphing with Points

• Finding Points:
  • Start at the origin (0,0)
  • Next points:
    • √1 = 1: (1,1)
    • √4 = 2: (4,2)
    • √9 = 3: (9,3)
• Vertical Stretch:
  • Example Function:
    • y = 2√x
    • Points from parent function double y-values
    • Translates to (1,2), (4,4), etc.

Example Function Analysis

• Function: y = √(x - 1) + 2
• New Origin: (1, 2)
  • Find points:
    • (1,2), (2,3), (5,4)
• Domain:
  • Lowest x value: 1
  • Domain: [1, ∞)
• Range:
  • Lowest y-value: 2
  • Range: 2, ∞)

Graphing Another Example Function

• Function: y = 3 - √(x - 4)
• Transformations:
  • Shifted 4 units right and 3 units up
• New Origin: (4, 3)
• Points:
  • (5, 2), (8, 1)
• Domain:
  • Lowest x-value: 4
  • Domain: [4, ∞)
• Range:
  • Highest y-value: 3
  • Range: (-∞, 3]

Conclusion

• Understanding transformations and shifts is crucial for graphing radical equations effectively.