Graphing Piecewise Functions

Overview

• Purpose: Learn to graph piecewise functions easily.
• Structure: Four problems of increasing difficulty.
  • Problem 1: Graphing two lines.
  • Problem 2: Includes a quadratic.
  • Problem 3: Graphing three different functions.
  • Problem 4: More complex with quadratics.
• Additional Resources: Link to printable notes and a supplementary video with further practice.

Problem 1: Two Lines

• Function 1: y = 5 when x <= -3
  • Graph is horizontal line at y=5.
  • Closed circle at x = -3 due to <=.
• Function 2: y = -2x - 3 for x > -3
  • Slope = -2, Y-intercept = -3.
  • Graph line using rise over run (down 2, over 1).
  • Open circle at x = -3 due to >.
• Result: Combined graph shows first part ends with closed circle, second part starts with open circle.

Problem 2: Line and Quadratic

• Function 1: y = 2x + 1 for x < 0
  • Slope = 2, Y-intercept = 1.
  • Graph using slope to determine line.
  • Open circle at x = 0 due to <.
• Function 2: y = x^2 - 3 for x >= 0
  • Y-intercept = -3.
  • Closed circle at x = 0 due to >=.
  • Plot points for x = 0, 1, 2, 3 and corresponding y-values.
• Result: Combination of a line and a parabola.

Problem 3: Three Functions

• Cutoffs at x = -2 and x = 2.
• Function 1: y = -4 for x <= -2
  • Horizontal line at y = -4.
  • Closed circle at x = -2.
• Function 2: y = x - 2 for -2 < x < 2
  • Slope = 1, Y-intercept = -2.
  • Open circle at x = 2, Closed circle at x = -2 (from previous function).
• Function 3: y = -2x + 4 for x >= 2
  • Slope = -2, Y-intercept = 4.
  • Closed circle at x = 2 due to >=.
• Result: Piecewise graph with distinct portions restricted by bounds.

Problem 4: Complex Piecewise

• Mark cutoffs at x = 0 and x = 5.
• Function 1: y = x^2 - 1 for x <= 0
  • Parabola starting from y-intercept -1.
  • Closed circle at x = 0.
  • Plot points for negative x-values and connect.
• Function 2: y = 2x - 1 for 0 < x <= 5
  • Slope = 2, Y-intercept = -1.
  • Closed circle at x = 5.
• Function 3: y = 3 for x > 5
  • Horizontal line at y=3.
  • Open circle at x = 5.
• Result: Complex graph with a parabola, line, and constant value section.

Additional Practice

• Suggested to practice with two-function piecewise graphs.
• Comment section activity: Identify open and closed circles by providing coordinates.

Extra Resources

• Link to an extra video for further practice and evaluation of piecewise functions.
• Printable notes available via provided link.