Lecture Notes: Graphing Radical Equations

Introduction

• Focus on graphing radical equations
• Start with basic parent function:
  • ( y = \sqrt{x} )

Parent Function: ( y = \sqrt{x} )

• Graph Characteristics:
  • Starts at the origin
  • Increases at a decreasing rate
• Domain: All non-negative x-values
  • Range: [0, ∞)
• Range: All non-negative y-values
  • Range: [0, ∞)

Transformations of ( y = \sqrt{x} )

Reflection

• Negative outside the radical:
  • Reflects over the x-axis
• Negative inside the radical:
  • Reflects over the y-axis
• Negative signs both inside and outside:
  • Reflects over the origin

Quadrants and Reflection

• Quadrants: Used to determine graph direction
  • Quadrant 1: (X, Y) both positive
  • Quadrant 2: (X negative, Y positive)
  • Quadrant 3: (X, Y) both negative
  • Quadrant 4: (X positive, Y negative)
• Use the sign of X (inside radical) and Y (outside radical) to determine direction:
  • Positive signs indicate movement towards Quadrant 1
  • Mixed signs indicate other quadrants

Vertical Shifts

• Adding/Subtracting outside radical:
  • +2: Shifts graph up 2 units
  • -1: Shifts down 1 unit

Horizontal Shifts

• Inside the radical:
  • ( \sqrt{x - 2} ): Shifts graph right by 2 units
  • ( \sqrt{x + 3} ): Shifts graph left by 3 units

Graphing Technique with Points

• Use specific points for accuracy:
  • Start at origin (0,0)
  • ( \sqrt{1} = 1 ): Move 1 right, 1 up
  • ( \sqrt{4} = 2 ): Move 4 right, 2 up
  • ( \sqrt{9} = 3 ): Move 9 right, 3 up
• Scaling:
  • If multiplied by 2, y-values double

Example: Graph ( \sqrt{x} - 1 + 2 )

• Domain: [1, ∞)
• Range: [2, ∞)
• Translation:
  • Start 1 unit right, 2 units up
  • Plot points: (1,2), (2,3), (5,4)

Additional Example

• Function: ( y = 3 - \sqrt{x - 4} )
  • Shift 4 units right, 3 units up
  • Reflects negatively over y-axis
• Domain: [4, ∞)
• Range: (-∞, 3]
• New Origin: (4, 3)
• Plotting Points:
  • Move 1 right and down 1 due to negative sign
  • Plot at (5,2), (8,1)

Conclusion

• Proper understanding of transformations and quadrant behavior aids in graphing radical equations accurately.
• Domains and ranges are critical for understanding the extent of the graph.