Functions and Modeling · Weeks 10-18

Linear vs. Non-Linear Functions

Comparing the properties of linear functions to functions that do not have a constant rate of change.

About This Topic

Understanding the distinction between linear and non-linear functions is a foundational concept for all subsequent algebra and modeling work. Linear functions produce a straight-line graph and have a constant rate of change. Non-linear functions have rates of change that vary, producing curves rather than lines. In 8th grade, students need to recognize this distinction from equations, graphs, and tables.

From an equation, the key indicator is whether the variable appears to the first power only. From a table, students check whether equal increments in x produce equal increments in y. From a graph, the question is whether the relationship produces a straight line. Each representation offers a different entry point, and fluency with all three is the goal.

Active sorting activities work especially well for this topic because classification tasks make students articulate their criteria. When a student has to defend why a parabola is non-linear to a skeptical partner or explain the difference between a curved graph and a straight line with a very steep slope, they build precision in their mathematical language that passive instruction rarely produces.

Learning Objectives

• Classify given functions as linear or non-linear based on their graphical representation.
• Calculate the rate of change from a table of values to determine if a function is linear.
• Compare the algebraic forms of linear and non-linear functions to identify key differences.
• Explain how a constant rate of change defines a linear function.
• Predict whether a real-world scenario, such as distance traveled at a constant speed versus the area of a growing square, will yield a linear or non-linear function.

Before You Start

Graphing Linear Equations

Why: Students need to be able to plot points and recognize a straight line on a coordinate plane.

Calculating Slope

Why: Understanding slope is essential for grasping the concept of a constant rate of change in linear functions.

Evaluating Functions

Why: Students must be able to substitute values into an equation to find corresponding outputs, a skill needed for creating tables of values.

Key Vocabulary

Linear Function	A function whose graph is a straight line. It has a constant rate of change.
Non-Linear Function	A function whose graph is not a straight line. Its rate of change varies.
Rate of Change	The measure of how much one quantity changes with respect to another quantity. For linear functions, this is constant.
Constant Rate of Change	The rate of change that remains the same between any two points of a function, indicating a linear relationship.

Active Learning Ideas

Sorting Activity: Linear or Not?

Provide groups with a set of cards showing functions as equations, graphs, and tables. Students sort them into linear and non-linear categories, then for each card explain which feature of that representation told them the answer. Sorting by representation type (all graphs together, all tables together) as a second pass deepens analysis.

⏱ 30 min·Small Groups

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Think-Pair-Share: Constant Rate Check

Give each pair a table of values with equal x-increments. Students calculate the differences in y-values, determine if the rate is constant, and classify the function. Pairs then predict what the graph of their function would look like before checking on graph paper or desmos.

⏱ 20 min·Pairs

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Real-World Prediction: Linear or Curve?

Present five real-world scenarios: simple interest earned over time, compound interest, constant speed, accelerating car, population growth at a fixed percentage. Students predict linear or non-linear for each, sketch a rough graph shape, and compare predictions in small groups before researching or computing to verify.

⏱ 25 min·Small Groups

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Real-World Connections

A taxi driver's fare structure often represents a linear function: a base fee plus a constant charge per mile traveled. This allows customers to easily predict the cost based on distance.

The growth of a plant measured by height over time is typically non-linear. Initial growth might be slow, then accelerate, and finally slow down again as the plant reaches maturity, creating a curved graph.

Engineers designing simple pulley systems might use linear functions to model the relationship between the force applied and the weight lifted, assuming ideal conditions with no friction.

Watch Out for These Misconceptions

Common Misconception

Students sometimes think that any graph that looks 'nearly straight' must be a linear function, especially when viewing a small portion of a curve.

What to Teach Instead

Show students the same function zoomed in versus zoomed out, or have them generate a table and check differences. Collaborative investigation of the constant-rate test is more convincing than a teacher demonstration alone.

Common Misconception

Students may believe that a steeper line is non-linear because it 'changes faster.'

What to Teach Instead

Reinforce that rate of change can be any constant value, including very large ones, and still produce a linear function. Pair activities comparing steep linear graphs to shallow curves make this distinction concrete.

Assessment Ideas

Quick Check

Provide students with a set of 5-7 cards, each displaying a different function's graph, equation, or table of values. Ask students to sort them into two piles: 'Linear' and 'Non-Linear', and be prepared to justify their placement of at least two cards.

Exit Ticket

Present students with two scenarios: 1) A car traveling at a steady 60 miles per hour. 2) The area of a square as its side length increases. Ask students to write one sentence explaining which scenario represents a linear function and why.

Discussion Prompt

Pose the question: 'If you have a table of values where the x-values increase by 1 each time, what must be true about the y-values for the function to be linear?' Facilitate a brief class discussion to solidify the concept of a constant rate of change.

Frequently Asked Questions

What active strategies help students distinguish linear from non-linear functions?

Sorting activities are particularly effective because they force students to state explicit criteria for classification rather than relying on visual guessing. Having students defend their sorting to a skeptical partner surfaces imprecise reasoning. Checking the constant-difference test on tables in pairs also builds reliable, replicable strategies for any representation.

How can you tell if a function is linear from a table of values?

Calculate the differences between consecutive y-values when the x-values increase by equal amounts. If those differences are constant, the function is linear. If the differences vary, the function is non-linear.

What makes a function non-linear according to CCSS 8.F.A.3?

A function is non-linear if it does not have a constant rate of change. This means its graph is not a straight line, its equation has a variable raised to a power other than one (or in a denominator or under a radical), and its table does not show equal output differences for equal input differences.

Why is recognizing non-linear functions important for 8th graders?

Real-world relationships are often non-linear: compound interest, area of a square as a side grows, population growth. Recognizing when linear models are insufficient prepares students to seek better-fit models and builds critical thinking about when simplifying assumptions break down.