Level Up Your Graphing: Piecewise Functions in Video Games

Materials Needed

• Paper or graph paper
• Pencil and colored pencils or highlighters
• Calculator, optional
• Ruler, optional
• Timer

Lesson Overview

Time: 15 minutes

Standard: MA.912.AR.9.10 — Solve and graph mathematical and real-world problems modeled with piecewise functions. Interpret key features and determine constraints in terms of the context.

Learning Target: I can graph piecewise functions.

Learning Objectives

By the end of the lesson, the learner will be able to:

• Identify the rule and domain for each part of a piecewise function.
• Graph each piece using open and closed circles correctly.
• Interpret key features, such as domain, range, slope, and endpoints, in a video-game context.

Success Criteria

I can:

• Graph each equation only over its stated interval.
• Use a closed circle for an included endpoint and an open circle for an excluded endpoint.
• Explain what the graph means in the video-game situation.

Introduction: Hook and Bell Work — 2 Minutes

Hook: Ask: “What might happen in a video game if the reward changes after a player reaches a certain level? Could one equation describe the reward for every level?”

Bell Work: Reinforce Prior Knowledge

Complete the following on paper:

1. Evaluate y = 3x + 2 when x = 2.
2. What is the slope of y = 3x + 2?
3. Plot the point (1, 4).

Quick answers: 1. 8; 2. 3; 3. Move right 1 and up 4.

Transition: “You already know how to evaluate and graph a linear equation. A piecewise function uses more than one linear rule, but each rule is used only during a specific part of the game.”

Body: I Do, We Do, You Do

I Do: Teacher or Parent Modeling — 4 Minutes

Use this video-game example:

Game reward function: A player earns coins based on the number of hours, h, played during a weekend event.

• For the first 2 hours, the player earns 100 coins per hour.
• After 2 hours and through 5 hours, the player earns 50 additional coins per hour.

Represent the situation with the piecewise function:

C(h) =
100h,    0 ≤ h ≤ 2
200 + 50(h − 2),    2 < h ≤ 5

Model these steps:

1. Read the restrictions. The first rule is used from 0 through 2 hours. The second rule is used after 2 hours through 5 hours.
2. Choose points for the first rule. For C(h) = 100h, use:
   • (0, 0)
   • (1, 100)
   • (2, 200)
3. Choose points for the second rule. For C(h) = 200 + 50(h − 2), use:
   • At h = 2, the output would be 200, but 2 is not included in this rule.
   • At h = 3, C(3) = 250.
   • At h = 5, C(5) = 350.
4. Graph only the allowed parts. Draw the first segment from 0 to 2 hours. Draw the second segment from just after 2 hours to 5 hours.
5. Use endpoint symbols. Use a closed circle when the endpoint is included and an open circle when it is not included.

Important graphing note: At (2, 200), the first rule has a closed circle because h = 2 is included. The second rule would have an open circle at the same point because its condition is 2 < h.

Interpretation:

• Domain: 0 ≤ h ≤ 5, because the event lasts no more than 5 hours.
• Range: 0 ≤ C(h) ≤ 350 coins.
• At 5 hours: The player earns 350 coins.
• Slopes: The first segment increases by 100 coins per hour; the second increases by 50 coins per hour.

We Do: Guided Practice — 4 Minutes

Work through this example together. Encourage the learner to explain each step aloud.

Game challenge points: A player earns points based on the number of completed challenges, x.

• For 0 through 3 challenges, the player earns 10 points per challenge.
• For more than 3 through 6 challenges, the player earns 20 points per challenge after the first 3 challenges.

P(x) =
10x,    0 ≤ x ≤ 3
30 + 20(x − 3),    3 < x ≤ 6

Complete together:

1. Make a table of values for each rule.
2. Identify the endpoint at x = 3.
3. Plot the points and connect only the points within each rule’s domain.
4. Determine the maximum number of points.

Answer: The maximum is 90 points at 6 completed challenges. The graph has a closed point at (3, 30) from the first rule and an open point at (3, 30) for the second rule.

You Do: Independent Application — 3 Minutes

Choose one of the following video-game challenges.

Option A: XP Boost

A player earns experience points, E(t), during a timed quest.

• For the first 4 minutes, the player earns 25 XP per minute.
• After 4 minutes and through 8 minutes, the player earns 40 XP per additional minute.

E(t) =
25t,    0 ≤ t ≤ 4
100 + 40(t − 4),    4 < t ≤ 8

Graph the function and answer:

1. What points should be plotted for the first rule?
2. What points should be plotted for the second rule?
3. Which endpoints are open or closed?
4. How much XP has the player earned after 8 minutes?

Option B: Create Your Own Game Function

Create a two-part piecewise function for a video-game situation, such as coins, XP, damage, health, score, or level rewards. Include:

• Two linear rules
• A restriction for each rule
• At least two points for each segment
• A labeled graph
• One sentence explaining the meaning of the endpoint and domain

Conclusion: Closure and Recap — 2 Minutes

Ask the learner to complete this verbal or written exit check:

1. A piecewise function uses one rule for the entire graph. True or false?
2. What does a closed circle mean?
3. Why must each piece be graphed only over its stated domain?
4. In a video-game context, what could the domain represent?

Expected responses:

• False. A piecewise function uses different rules over different intervals.
• A closed circle means the endpoint is included.
• The restrictions describe when that rule applies in the real-world situation.
• The domain could represent time played, number of challenges, player level, or another input.

Final takeaway: “To graph a piecewise function, graph each rule only where it is allowed, use open and closed circles correctly, and explain what the graph means in the context.”

Assessment

Formative Assessment

• Check bell-work responses for understanding of evaluating linear functions, slope, and plotting points.
• Ask the learner to explain why each endpoint is open or closed.
• Observe whether the learner graphs only within each stated interval.
• Use the guided-practice table to check calculations before independent work.

Summative Assessment

The learner successfully completes the independent graph if the response includes:

• Correctly calculated points: 2 points
• Correctly graphed line segments: 2 points
• Correct open and closed circles: 2 points
• Correct domain and contextual interpretation: 2 points
• Accurate final answer and clear explanation: 2 points

Differentiation and Adaptations

• Support: Provide a partially completed table, label the axes, and use colored pencils to show each rule in a different color.
• Additional support: Begin with whole-number inputs and use a calculator for evaluating the rules.
• Auditory option: Have the learner explain each graphing step aloud or record a brief explanation.
• Digital option: Use an online graphing calculator or spreadsheet to plot the points and compare the graph with the hand-drawn version.
• Extension: Create a piecewise function with a jump, such as a game reward that suddenly increases at a new level. Explain how the jump appears on the graph.
• Real-world connection: Replace game points with a household situation, such as chore rewards, workout minutes, savings, or delivery fees.