Piecewise Functions and Writing Equations

Learning Objective

• Write the equation of a piecewise function from a graph.

---

Example 1: Writing the Equation of a Piecewise Function

Functions Identified

1. First function (Red):

   • Type: Line
   • Equation:
     • Slope (m): -1 (rise/run of -1/1)
     • Y-intercept (b): 3
   • Final Equation:
     • y = -1x + 3

2. Second function (Green):

   • Type: Line
   • Equation:
     • Slope (m): -2 (rise/run of -2/1)
     • Y-intercept: Found at 3
   • Final Equation:
     • y = -2x + 3

Piecewise Notation

• y =
  • -1x + 3 for x ≤ -2
  • -2x + 3 for x > 2

---

Useful Tools for Piecewise Functions

• Identifying Domains:
  • Vertical lines: x = number
  • Horizontal lines: y = number

General Information on Translating Functions

• Linear and Absolute Value Functions
  • Remember the format to identify slopes and intercepts.

---

Example 2: Writing the Equations of Three Functions

Functions Identified

1. First function (Red):

   • Type: Line
   • Slope (m): -1/2
   • Equation:
     • y = -1/2(x + 1) - 1

2. Second function (Green):

   • Type: Horizontal Line
   • Equation:
     • y = 2

3. Third function (Blue):

   • Type: Line
   • Slope (m): -1
   • Equation:
     • y = -1(x - 3) + 4

Domains

• Red function:

  • x < -1 (open circle)

• Green function:

  • -1 ≤ x ≤ 3 (includes both)

• Blue function:

  • x > 3 (open circle)

---

Example 3: Writing Equations of Three More Functions

Functions Identified

1. First function (Blue):

   • Type: Absolute Value
   • Equation:
     • y = 2|x + 3| + 2

2. Second function (Red):

   • Type: Horizontal Line
   • Equation:
     • y = 4

3. Third function (Green):

   • Type: Line
   • Slope (m): -3
   • Equation:
     • y = -3(x - 3) + 4

Domains

• Blue function:

  • x ≤ -2 (includes)

• Red function:

  • -2 < x < 3 (not including)

• Green function:

  • x ≥ 3 (includes)

---

Summary

• Ensure to write down all examples and useful tools in your notes for future reference.
• Practice recognizing different types of functions and their equations.