Mathematics · 9th Grade · The Language of Algebra · Weeks 1-9

Analyzing Graphs of Functions

Interpreting key features of function graphs, including intercepts, intervals of increase/decrease, and end behavior.

• Common Core State Standards
• CCSS.Math.Content.HSF.IF.B.4
• CCSS.Math.Content.HSF.IF.C.7

About This Topic

Reading a function graph is one of the most practical skills in U.S. high school mathematics, connecting algebraic notation to visual information that appears in science, economics, and statistics. In the Common Core framework, students at this level analyze key features including x-intercepts, y-intercepts, intervals of increase and decrease, and end behavior. Each feature carries specific meaning about the function's behavior, and students who can extract that meaning from a graph are better equipped for the modeling work that runs through high school and beyond.

Intervals of increase and decrease communicate how the function's output responds to changes in input. End behavior describes what happens to the output as the input grows very large or very small. These ideas extend far beyond graphs of lines, giving students a vocabulary and framework for all function families they will encounter in algebra and later courses.

Active learning strategies that ask students to narrate a graph as if it tells a story are especially productive here. When students describe a graph in terms of a real-world context, they are forced to connect the visual features to their functional meaning, which is exactly the kind of interpretation the standards require.

Key Questions

1. Explain how to identify the x and y-intercepts from a function's graph.
2. Analyze what the intervals of increasing and decreasing tell us about a function's behavior.
3. Predict the end behavior of a function based on its graphical representation.

Learning Objectives

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

Before You Start

Plotting Points and Coordinate Plane

Why: Students need to be able to accurately plot points and understand the structure of the coordinate plane to interpret graphical representations of functions.

Introduction to Functions

Why: Understanding the concept of a function, including input-output relationships and function notation, is essential before analyzing their graphical features.

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).

Watch Out for These Misconceptions

Common Misconception

Students confuse x-intercepts with y-intercepts, often labeling both incorrectly or describing them only as 'where it crosses the axis' without specifying which axis.

What to Teach Instead

Establish a consistent verbal habit: the y-intercept is where x = 0 (the function's starting output), and the x-intercept is where y = 0 (where the output equals zero). Color-coding axes during graph analysis activities helps anchor this.

Common Misconception

Students describe intervals of increase or decrease using y-values instead of x-values (e.g., 'the function increases from 2 to 7' when 2 and 7 are y-values).

What to Teach Instead

Reinforce that intervals are always stated in terms of x, the input. Use the phrasing: 'As x goes from ___ to ___, y is increasing/decreasing.' Graph narrative activities where students explain in words require them to be explicit about which axis drives the description.

Active Learning Ideas

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.

25 min·Individual

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.

20 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.

35 min·Pairs

Real-World Connections

• Economists analyze graphs of supply and demand curves to identify equilibrium points (intercepts) and understand how prices change (intervals of increase/decrease) as production levels vary.
• Engineers use graphs of stress-strain relationships to determine the yield strength (an intercept) and the elastic region (intervals of increase) before a material permanently deforms.
• Biologists interpret graphs showing population growth or decline over time, noting when populations reach carrying capacity (intercepts) and periods of rapid growth or decline (intervals of increase/decrease).

Assessment Ideas

Exit Ticket

Provide students with a printed graph of a non-linear function. Ask them to: 1. Identify and label the y-intercept. 2. State one interval where the function is increasing. 3. Describe the end behavior as x approaches positive infinity.

Quick Check

Display three different graphs on the board. For each graph, ask students to hold up fingers to indicate: 1 finger for 'one x-intercept', 2 fingers for 'two x-intercepts', 0 fingers for 'no x-intercepts'. Repeat for intervals of increase/decrease and end behavior descriptions.

Discussion Prompt

Present a scenario, such as 'The temperature in a city over a 24-hour period.' Ask students: 'How would you represent this on a graph? What would the y-intercept represent? What would an interval of decrease tell us about the temperature?'

Frequently Asked Questions

How do you find the x-intercepts and y-intercepts of a function from its graph?

The y-intercept is the point where the graph crosses the vertical axis. It occurs at x = 0 and is read directly from the graph as the y-coordinate of that crossing point. The x-intercept is where the graph crosses the horizontal axis, where y = 0. A function may have zero, one, or multiple x-intercepts but only one y-intercept.

What does it mean for a function to be increasing or decreasing?

A function is increasing on an interval when the output (y-value) gets larger as the input (x-value) increases, moving left to right on the graph. It is decreasing on an interval when the output gets smaller as the input increases. These intervals are always described using x-values, such as 'decreasing on x from -1 to 3.'

What is end behavior in a function graph?

End behavior describes the direction the function's output heads as the input grows toward positive infinity (far right) or negative infinity (far left). For example, a line with positive slope rises to the right and falls to the left. In 9th grade, end behavior is described verbally from a graph rather than with formal limit notation, which comes in calculus.

How does active learning help students interpret function graphs?

Writing graph narratives in a real-world context requires students to assign meaning to every key feature rather than just listing them. When students must explain that 'the x-intercept at week 6 means the project was back on schedule,' they connect the mathematical feature to its interpretation. Matching activities add the additional challenge of reading descriptions and verifying them against graphs, reinforcing both directions of translation.