Analyzing Graphs of Functions

Graph analysis becomes meaningful when students move from passive observation to active interpretation, especially for functions. Drawing out the story behind the curve helps students connect abstract notation to real-world events, reinforcing both algebraic fluency and modeling skills.

• 9th Grade
• Mathematics
• 3 activities
• 20 min–35 min

Learning Objectives

1. Identify the x-intercept(s) and y-intercept of a function from its graph and explain their meaning in context.
2. Analyze intervals where a function's graph is increasing or decreasing, and describe the corresponding change in the function's output.
3. Predict the end behavior of a function by examining its graph as the input approaches positive or negative infinity.
4. Compare the graphical representations of different functions to determine similarities and differences in their intercepts, intervals of increase/decrease, and end behavior.

Gallery Walk

⏱ 25 min·Individual

Graph Narrative: Tell the Story

Provide each student with a graph of a function representing a real-world scenario (distance driven, account balance, water temperature). Students write a three-paragraph narrative describing the journey of the input and output, explicitly naming intercepts, increasing and decreasing intervals, and what happens at the far left and far right of the graph.

Prepare & details

Explain how to identify the x and y-intercepts from a function's graph.

Facilitation Tip: During Graph Narrative: Tell the Story, have students highlight the y-axis in one color and the x-axis in another to prevent axis confusion when labeling intercepts.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Feature Hunt

Project a function graph without context. Students individually list every key feature they can identify with its location (e.g., y-intercept at (0, 3), decreasing on the interval from x = 2 to x = 5). Partners compare lists, debate any discrepancies, and together write the complete feature list to share with the class.

Prepare & details

Analyze what the intervals of increasing and decreasing tell us about a function's behavior.

Facilitation Tip: During Think-Pair-Share: Feature Hunt, circulate and listen for students using phrases like 'as x goes from… to…' to ensure intervals are described in terms of input values.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Gallery Walk

⏱ 35 min·Pairs

Matching Activity: Graph to Feature Description

Prepare eight graphs and eight written feature descriptions. Students work in pairs to match each graph to its description, justifying each match in one sentence. Two distractors (descriptions that fit no graph) are included to force careful reading. Pairs then create their own graph and write its feature description for another pair to match.

Prepare & details

Predict the end behavior of a function based on its graphical representation.

Facilitation Tip: During Matching Activity: Graph to Feature Description, provide a reference card showing axis colors and feature labels to anchor student discussions.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Start by modeling how to read a graph aloud, pausing at each feature and naming it by its definition. Avoid rushing to symbolic notation; instead, build the habit of translating visuals into spoken descriptions first. Research shows that students benefit from color-coding and consistent verbal frames, such as 'when x is zero, the y-value is…' for y-intercepts.

What to Expect

Successful learners will articulate how each graph feature reflects the function’s behavior, using precise language tied to the coordinate plane. They will distinguish intercepts by axis, describe intervals with correct input ranges, and explain end behavior in terms of x approaching infinity.

Watch Out for These Misconceptions

Common Misconception

During Graph Narrative: Tell the Story, watch for students confusing which axis corresponds to each intercept, often describing the y-intercept as 'where it crosses the x-axis'.

What to Teach Instead

Ask students to verbally define each intercept before labeling: 'The y-intercept is where the function meets the y-axis, which happens when x equals zero. Please circle that point and label it y-intercept.'

Common Misconception

During Think-Pair-Share: Feature Hunt, watch for students describing intervals in terms of y-values, such as 'the function increases from 2 to 7'.

What to Teach Instead

Prompt students to restate their interval using x: 'When you say the function increases from 2 to 7, are 2 and 7 x-values or y-values? Please rephrase using x-values from the graph.'

Assessment Ideas

Exit Ticket

After the exit-ticket activity, review student responses to ensure they correctly identify the y-intercept, state an increasing interval using x-values, and describe end behavior with x approaching infinity.

Quick Check

During Matching Activity: Graph to Feature Description, circulate and listen to student pairs justify their matches, focusing on whether they use correct terminology for intercepts and intervals.

Discussion Prompt

After Graph Narrative: Tell the Story, have students share their narratives in small groups and ask peers to point out any features that were mislabeled or described incorrectly.

Extensions & Scaffolding

• Challenge: Ask students to sketch a new function graph that meets a set of constraints (e.g., two x-intercepts, increasing on 0 < x < 3, end behavior y approaches 2 as x approaches negative infinity).
• Scaffolding: Provide a partially completed graph with labeled x- and y-axes, and ask students to fill in missing intercepts and intervals before writing their narrative.
• Deeper exploration: Introduce piecewise functions and ask students to create a narrative that explains how different pieces connect at domain boundaries.

Key Vocabulary

x-intercept	A point where the graph of a function crosses or touches the x-axis. At these points, the y-coordinate is always zero.
y-intercept	A point where the graph of a function crosses or touches the y-axis. At this point, the x-coordinate is always zero.
interval of increase	A range of x-values for which the function's output (y-values) increases as the input (x-values) increases.
interval of decrease	A range of x-values for which the function's output (y-values) decreases as the input (x-values) increases.
end behavior	Describes what happens to the y-values of a function as the x-values approach positive infinity (very large positive numbers) or negative infinity (very large negative numbers).