Graph Detective: Finding Increasing, Decreasing, Positive, and Negative Functions

Materials Needed

• Paper or graph paper
• Pencil and colored pencils or highlighters
• Ruler
• Timer
• Optional: graphing calculator or digital graphing tool
• Prepared graph cards or the sample graphs described below

Learning Objectives

By the end of this 20-minute lesson, the learner will be able to:

• Identify intervals where a function is increasing or decreasing by reading a graph from left to right.
• Identify intervals where a function is positive or negative by locating the graph above or below the x-axis.
• Describe answers using interval notation or clear inequality statements.
• Explain how a graph’s location and direction provide information about a function.

Success Criteria

I can:

• Say, “The graph is increasing when it rises from left to right.”
• Say, “The graph is decreasing when it falls from left to right.”
• Find where the graph is positive by identifying where it is above the x-axis.
• Find where the graph is negative by identifying where it is below the x-axis.
• Use correct interval endpoints and explain what happens at zeros.

Key Vocabulary

• Increasing: The graph rises as you move from left to right.
• Decreasing: The graph falls as you move from left to right.
• Positive: Function values are greater than zero; the graph is above the x-axis.
• Negative: Function values are less than zero; the graph is below the x-axis.
• Zero: A point where the graph touches or crosses the x-axis.

Introduction: Hook and Purpose — 2 Minutes

Hook: Imagine you are tracking a roller coaster. The track rises, falls, and sometimes crosses ground level. A graph tells the same story:

• Rising track = increasing function
• Falling track = decreasing function
• Above ground level = positive function
• Below ground level = negative function

Tell the learner: “Today you will become a graph detective. You will investigate the direction of a graph and determine whether its output is positive or negative.”

Body: I Do — Teacher/Parent Modeling — 5 Minutes

1. Model Increasing and Decreasing

Draw or display a graph with a curve that:

• Rises from the left until x = 2
• Falls from x = 2 until x = 5
• Rises again after x = 5

Think aloud:

1. “I read a graph from left to right, just as I read a sentence.”
2. “From the far left to x = 2, the graph rises, so the function is increasing on the interval before 2.”
3. “From x = 2 to x = 5, the graph falls, so the function is decreasing on (2, 5).”
4. “After x = 5, the graph rises again, so the function is increasing after 5.”

Memory cue: Move your finger from left to right. If your finger moves up, the function is increasing. If it moves down, the function is decreasing.

2. Model Positive and Negative

Use the same graph and draw a bold horizontal x-axis.

1. “The function is positive wherever the graph is above the x-axis.”
2. “The function is negative wherever the graph is below the x-axis.”
3. “Where the graph crosses the x-axis, the function equals zero. Those points are not included in the positive or negative intervals.”

Color the part above the x-axis green and the part below it red. Explain that positive and negative describe the output, or y-value, not the direction of the graph.

Body: We Do — Guided Practice — 5 Minutes

Draw or display a simple graph with these features:

• The graph decreases from x = −5 to x = −2.
• The graph increases from x = −2 to x = 1.
• The graph decreases from x = 1 to x = 4.
• The graph crosses the x-axis at x = −4 and x = 3.

Work together through the following questions. Ask the learner to point to the graph and explain each answer.

1. Where is the graph decreasing?

   Expected answer: The graph is decreasing on (−5, −2) and (1, 4).

2. Where is the graph increasing?

   Expected answer: The graph is increasing on (−2, 1).

3. Where is the function positive?

   Identify the portions above the x-axis. The exact answer depends on the drawn graph. Have the learner use the intercepts at −4 and 3 to mark the intervals.

4. Where is the function negative?

   Identify the portions below the x-axis. Remind the learner to exclude the zeros at x = −4 and x = 3.

Quick check: Ask, “Can a function be increasing while it is negative?”

Expected response: Yes. A graph can rise while remaining below the x-axis. Increasing/decreasing describes direction; positive/negative describes location compared with the x-axis.

Body: You Do — Independent Graph Detective Challenge — 5 Minutes

Give the learner this scenario or have the learner sketch a graph that matches it:

A function has the following behavior:

• It decreases on (−6, −3).
• It increases on (−3, 2).
• It decreases on (2, 6).
• It is above the x-axis on (−5, 0) and (4, 6).
• It is below the x-axis on (−6, −5) and (0, 4).

Ask the learner to complete the following independently:

1. Circle the intervals where the function is increasing.
2. Underline the intervals where the function is decreasing.
3. Shade the intervals where the function is positive.
4. Place an “N” beside the intervals where the function is negative.
5. Explain in one sentence how increasing differs from positive.

Expected answers:

• Increasing: (−3, 2)
• Decreasing: (−6, −3) and (2, 6)
• Positive: (−5, 0) and (4, 6)
• Negative: (−6, −5) and (0, 4)
• Explanation: Increasing describes whether the graph rises from left to right, while positive describes whether the graph is above the x-axis.

Conclusion: Closure and Recap — 3 Minutes

Ask the learner to complete these statements aloud:

• “A graph is increasing when it __________ from left to right.”
• “A graph is decreasing when it __________ from left to right.”
• “A function is positive when its graph is __________ the x-axis.”
• “A function is negative when its graph is __________ the x-axis.”

Exit Ticket: On a small piece of paper, draw a graph that is:

1. Increasing but negative
2. Decreasing but positive

Have the learner label each graph and explain why it meets the description.

Assessment

Formative Assessment

• Listen for whether the learner reads the graph from left to right.
• Ask the learner to point to evidence on the graph for every answer.
• Use the quick-check question to distinguish direction from location.
• Correct errors immediately by asking, “Are we describing the graph’s direction or its position relative to the x-axis?”

Summative Assessment

The learner demonstrates mastery by correctly identifying at least 4 out of 5 features on the independent challenge and accurately completing the exit ticket.

Differentiation and Flexibility

Support for a Learner Who Needs More Scaffolding

• Use a finger, ruler, or pencil to trace the graph from left to right.
• Use arrows to mark rising and falling sections.
• Color above the x-axis green and below the x-axis red.
• Begin with whole-number intervals before introducing interval notation.
• Provide sentence frames: “The function is increasing on ___ because the graph ___.”

Extension for an Advanced Learner

• Use a graphing tool to create a piecewise function with at least two increasing and two decreasing intervals.
• Label its positive and negative intervals using interval notation.
• Explain how a local maximum or local minimum affects increasing and decreasing intervals.
• Write a real-world story, such as temperature, elevation, or bank balance, that matches the graph.

Real-World Connection

Explain that these skills are used to interpret temperatures, speed, profit, population, height, and water levels. For example, a business graph may be increasing while profit is still negative: profit is improving, but the business has not yet reached zero profit.