Algebra Lesson: Piecewise and Step Functions

Objectives

• By the end of this lesson, students will be able to:
  • Graph piecewise functions and identify their domain and range.
  • Graph step functions and identify their domain and range.

Piecewise Functions

• Definition: A piecewise function is defined by different equations over different intervals of its domain.
• Graphing Method:
  • Use an x-y table to plot points.
  • For each piece, determine the applicable x-values and calculate corresponding y-values.
  • Plot points and connect them, ensuring arrows indicate the direction of the graph.

Example 1

1. Function 1: For x ≤ 4, use the equation x + 3.
   • Calculate y for x-values 4, 3, 2.
   • Plot these on a graph.
   • Connect with an arrow indicating direction.
2. Function 2: For x > 4, use the equation (-\frac{3}{2}x + 6).
   • Select x-values such as 4, 6, 8 to match denominator.
   • Calculate and plot these points, leaving the point at x=4 as an open circle.
   • Connect and indicate direction with an arrow.

Domain and Range

• Domain: For piecewise functions often all real numbers.
• Range: Determined by the highest and lowest y-values on the graph.
  • Example 1: Range is y ≤ 5.

Step Functions

• Definition: A specific type of piecewise function with constant values over intervals, resembling steps.
• Greatest Integer Function: Rounds down to the nearest integer.
  • Use fancy brackets to denote rounding.
  • Graphing involves marking steps with filled circles on the left and open circles on the right.

Example 2

• Values Inside Brackets:

  • Compute rounded integer values and construct a table with x-values including decimals.
  • Graph these as steps.

• Domain and Range:

  • Domain for step functions is all real numbers.
  • Range includes only integers (no decimals), often described as multiples of a number defining the step size.

Example 3

• Values Outside Brackets: Affects step height.
  • Multiply rounded values by a factor like negative 3.
  • Steps are larger, e.g., moving by multiples of 3.

Identifying Domain and Range

• For all examples, the domain is all real numbers.
• The range is determined by integers or multiples of a specific factor (e.g., 3, 7).

Practice

• Students are encouraged to create tables, graph functions, and identify domains and ranges for given equations.

Conclusion

• If there are questions or clarifications needed, students are encouraged to reach out.