Graphing Absolute Value Function: ( f(x) = \frac{1}{2} |x + 2| - 3 )

Key Concepts

• Absolute Value Function: Generally takes the shape of a 'V'.
  • Example: ( g(x) = |x| )
  • All absolute value functions will form a V shape, possibly wider, narrower, opening up or down.
• Vertex: The highest or lowest point on the graph of the absolute value function.
  • Finding the vertex is crucial for graphing.

Graphing Steps

1. Identify the Vertex:

   • Set the expression inside the absolute value to zero. In this case, solve ( x + 2 = 0 ).
     • Solution: ( x = -2 )
   • This makes the vertex at point ((-2, -3)).

2. Choose Additional Points

   • Select one ( x ) value to the left of the vertex and one to the right.
   • Use multiples of 2 inside the absolute value to avoid fractions:
     • For ( x = 0 ), calculate ( f(0) = \frac{1}{2} \times |0 + 2| - 3 = -2 ). Point: ((0, -2)).
     • For ( x = -4 ), calculate ( f(-4) = \frac{1}{2} \times |-4 + 2| - 3 = -2 ). Point: ((-4, -2)).

3. Plot and Draw Graph

   • Plot the points ((-2, -3), (0, -2), (-4, -2)) on the graph.
   • Draw lines forming a V shape through these points.

Transformations

• Shift and Compress:
  • The function ( f(x) = \frac{1}{2} |x + 2| - 3 ) can also be graphed using transformations:
    • Horizontal Shift: Left by 2 units due to ( x + 2 ).
    • Vertical Compression: Multiply all ( y )-coordinates by ( \frac{1}{2} ).
    • Vertical Shift: Down by 3 units due to (-3).

Conclusion

• Understanding these transformations and plotting steps helps in graphing any variation of the absolute value function efficiently.