Graphing Piecewise Functions Tutorial

Example 1: f(x) = { x, x < 0; 5, x ≥ 0 }

• Graph f(x) = x or y = x:

  • Straight line with a slope of 1
  • Graph rises at a 45-degree angle

• Graph f(x) = 5 or y = 5:

  • Horizontal line at y = 5

• Piecewise Graph:

  • For x < 0, graph y = x with an open circle at (0,0) (excludes 0)
  • For x ≥ 0, graph y = 5 with a closed circle at (0,5) (includes 0)

Example 2: f(x) = { 2, x < 1; x + 3, x > 2 }

• Graph f(x) = 2:

  • Horizontal line at y = 2 for x < 1
  • Open circle at (1,2) (excludes 1)

• Graph f(x) = x + 3:

  • Linear graph with y-intercept 3, slope 1
  • Start at (2,5), open circle (since x > 2)
  • Increases at a 45-degree angle

Example 3: f(x) = { 2x + 1, x < 1; 1, x = 1; -x^2, x > 1 }

• Graph 2x + 1:

  • Linear graph with slope 2, y-intercept 1
  • Open circle at (1,3) (excludes 1)
  • For x = 0, y = 1

• Point at x = 1:

  • Closed circle at (1,1)

• Graph -x^2:

  • Downward opening parabola
  • Open circle at (1,-1)
  • Point at (2,-4)

Example 4: f(x) = { 3x + 4, x < 0; 2, x = 0; √x, x > 1 }

• Graph 3x + 4:

  • Linear graph with slope 3, y-intercept 4
  • Open circle at (0,4)
  • Point at (-1,1) using slope

• Point at x = 0:

  • Closed circle at (0,2)

• Graph √x:

  • Increasing function
  • Open circle at (1,1)
  • Point at (4,2)

Example 5: f(x) = { 1/x, x < 0; 3, 0 < x < 3; -x + 5, x ≥ 3 }

• Graph 1/x:

  • Hyperbola
  • Only negative side (x < 0)

• Graph y = 3:

  • Horizontal line between x = 0 and x = 3
  • Closed circle at (0,3), open circle at (3,3)

• Graph -x + 5:

  • Linear with slope -1
  • Closed circle at (3,2)
  • Points like (4,1) and (5,0) (x-intercept)

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• Key Points:
  • Piecewise functions are graphed segment by segment.
  • Open vs closed circles indicate excluded vs included endpoints.
  • Be mindful of the slope and intercept when sketching linear parts.
  • Check for continuity at endpoints when relevant.