I Can Graph Piecewise Functions
Materials Needed
- Graph paper or a digital graphing tool
- Pencil and colored pencils or highlighters
- Ruler
- Calculator, optional
- Exit ticket, included below
Learning Objectives
By the end of this 20-minute lesson, the learner will be able to:
- Identify the rule and domain for each part of a piecewise function.
- Graph each rule only on its stated domain.
- Use open and closed circles correctly to show whether endpoints are included.
- Explain how the graph represents a real-world situation that changes based on conditions.
Success Criteria
I can say, “I am successful when I can…”
- Read each condition, such as x < 2 or x ≥ 2.
- Graph each equation on the correct part of the coordinate plane.
- Use a closed circle for an included endpoint and an open circle for an excluded endpoint.
- Check that my graph has no points outside the stated domains.
Introduction: Hook and Objectives — 2 Minutes
Hook: Imagine a taxi company that charges one price for the first few miles and a different price after that. Would one equation describe the entire cost? Probably not! A rule that changes depending on the situation is called a piecewise function.
Today, you will learn how to graph a piecewise function by treating each rule as a separate “graphing mission.” You will first watch a model, then solve one with guidance, and finally complete one independently.
Mini-Lesson: What to Look For — 2 Minutes
A piecewise function has:
- A rule: the equation to graph, such as x + 1.
- A condition: the domain where that rule applies, such as x < 2.
- An endpoint symbol:
- Closed circle: the endpoint is included, using ≤ or ≥.
- Open circle: the endpoint is not included, using < or >.
Graphing routine: Read the condition, choose valid x-values, calculate y-values, graph the points, draw the portion of the line, and mark the endpoint correctly.
Body: I Do — Teacher/Parent Model — 5 Minutes
Model the following example aloud:
Example:
f(x) = x + 1, when x < 2
f(x) = 4, when x ≥ 2
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Graph the first piece: y = x + 1 when x < 2.
- Choose x-values less than 2, such as −1, 0, and 1.
- The points are (−1, 0), (0, 1), and (1, 2).
- Because x = 2 is not included, calculate the endpoint: 2 + 1 = 3.
- Place an open circle at (2, 3).
- Draw the line only to the left of x = 2.
-
Graph the second piece: y = 4 when x ≥ 2.
- This is a horizontal line at y = 4.
- Because x = 2 is included, place a closed circle at (2, 4).
- Draw the horizontal line to the right from x = 2.
Think aloud: “The two pieces have different rules, so I graph them separately. The condition tells me where each piece is allowed to appear. The symbols tell me whether the endpoint is open or closed.”
Quick check: Ask the learner: “Why is the circle at (2, 3) open, while the circle at (2, 4) is closed?”
Body: We Do — Guided Practice — 4 Minutes
Work through this example together. Encourage the learner to explain each step before graphing.
Guided Example:
g(x) = 2x, when x ≤ 1
g(x) = x + 1, when x > 1
Use these guiding questions:
- What is the rule for the first piece?
- Which x-values may be used for the first piece?
- Is x = 1 included in the first piece? Which circle should be used?
- What is the endpoint for the first rule? 2(1) = 2, so the endpoint is (1, 2).
- What is the rule for the second piece?
- Is x = 1 included in the second piece? Which circle should be used?
- What is the endpoint for the second rule? 1 + 1 = 2, so use an open circle at (1, 2).
Have the learner plot at least two points for each piece:
- For y = 2x when x ≤ 1: use x = −1, 0, and 1.
- For y = x + 1 when x > 1: use x = 2, 3, and 4.
Formative assessment: The learner should explain why the first piece has a closed circle and the second piece has an open circle, even though both pieces meet at the same coordinate.
Body: You Do — Independent “Graphing Mission” — 5 Minutes
Complete the following independently. Choose graph paper or a digital graphing tool.
Challenge:
h(x) = −x + 2, when x < 3
h(x) = 5, when x ≥ 3
Instructions:
- Underline the condition for each piece.
- Make a small table of at least three points for each rule.
- Graph the first rule only for x-values less than 3.
- Graph the second rule only for x-values greater than or equal to 3.
- Mark the endpoint for each piece with an open or closed circle.
- Use a different color for each piece if possible.
Optional real-world connection: Pretend this function represents a game score. Before level 3, the score follows the first rule; starting at level 3, the score changes to a constant value. Write one sentence explaining what the change in rule means.
Conclusion: Closure and Exit Ticket — 2 Minutes
Ask the learner to complete this exit ticket without looking at the examples:
- What does the condition in a piecewise function tell you?
- When do you use an open circle?
- When do you use a closed circle?
- Describe one step you use to graph a piecewise function.
Recap: To graph a piecewise function, graph each rule separately, follow only the domain listed for that rule, and use open or closed circles to show whether the endpoint is included.
Answer Check for the Independent Task
- For y = −x + 2 when x < 3, the endpoint is (3, −1) with an open circle. Draw the line to the left.
- For y = 5 when x ≥ 3, the endpoint is (3, 5) with a closed circle. Draw a horizontal line to the right.
Assessment
- Formative: Responses during the “I Do” quick check and guided practice.
- Summative: Completed independent graph and exit ticket.
- Proficiency target: Correctly graph both pieces, use both endpoint symbols accurately, and follow each domain with at least 80% accuracy.
Differentiation and Choice
- Additional support: Provide a step-by-step checklist, allow a table of values, and highlight each condition in a different color.
- Visual support: Shade the allowed domain on the x-axis before graphing each piece.
- Verbal support: Have the learner say, “Rule, domain, points, endpoint, graph” while working.
- Extension: Create a piecewise function with two different linear rules, graph it, and write a real-world story that matches the conditions.
- Digital option: Use a graphing calculator or graphing website to check the hand-drawn graph after completing it independently.