Linear Functions and Graphs Review

Students need to move beyond memorizing graph shapes to truly understand how different functions behave. Active learning helps them connect the abstract rules of equations to the concrete visuals of graphs, making it easier to spot patterns and avoid confusion between function types.

• Secondary 3
• Mathematics
• 3 activities
• 20 min–50 min

Learning Objectives

1. Analyze how changes in the gradient and y-intercept of a linear function affect the position and steepness of its graph.
2. Compare the efficiency of using the slope-intercept form (y=mx+c) versus the standard form (ax+by=c) to solve specific linear equation problems.
3. Calculate the gradient and y-intercept for a given linear equation in any form.
4. Explain the relationship between the algebraic representation of a linear function and its graphical representation.
5. Predict the intersection point of two linear graphs given their equations.

Stations Rotation

⏱ 50 min·Small Groups

Stations Rotation: The Function Zoo

Set up stations for Cubic, Reciprocal, and Exponential functions. At each station, students must plot a few points, identify the shape, and find one unique feature (like an asymptote or a point of inflection) before moving to the next.

Prepare & details

Analyze how the gradient and y-intercept define a linear function's graph.

Facilitation Tip: During Station Rotation: The Function Zoo, position one station with graphing software so students can adjust parameters and immediately see how changes affect the curve.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Inquiry Circle

⏱ 35 min·Pairs

Inquiry Circle: The Growth Race

Give pairs three functions: linear, quadratic, and exponential. Students calculate values for x=1, 2, 5, 10 and 20. They then discuss and present which function 'wins' the race in the long run and why the exponential curve eventually overtakes the others.

Prepare & details

Compare the advantages of using different forms of linear equations.

Facilitation Tip: For Collaborative Investigation: The Growth Race, provide a mix of graph paper and digital tools so students can choose their preferred method for analyzing growth patterns.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Asymptote Mystery

Show a graph of y = 1/x and ask students what happens as x gets closer and closer to zero. After individual thinking and pairing, the class discusses why the calculator gives an 'error' and how that translates to the gap in the graph.

Prepare & details

Predict the behavior of a linear graph given changes in its gradient or y-intercept.

Facilitation Tip: In Think-Pair-Share: The Asymptote Mystery, give pairs a whiteboard to sketch and label the behavior of the graph as it approaches the asymptote.

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

Teaching This Topic

Start by connecting new function types to familiar linear functions to build on prior knowledge. Use real-world examples to make abstract concepts tangible, such as comparing cubic functions to volume calculations or exponential functions to interest rates. Avoid rushing through the material—give students time to explore each graph type thoroughly before moving to the next.

What to Expect

By the end of these activities, students should confidently identify and distinguish cubic, reciprocal, and exponential graphs by their key features. They should also explain why certain graphs have asymptotes or specific turning points, using mathematical language to support their reasoning.

Watch Out for These Misconceptions

Common Misconception

During Station Rotation: The Function Zoo, watch for students who assume an exponential decay curve will eventually reach zero on the graph.

What to Teach Instead

Ask students to use a calculator to evaluate values like 0.5 to the power of 10 or 0.5 to the power of 20, then plot these points to see the curve’s approach toward zero without touching it.

Common Misconception

During Collaborative Investigation: The Growth Race, watch for students who confuse the smooth 'S' shape of a cubic graph with the separate branches of a reciprocal graph.

What to Teach Instead

Have students physically trace each curve with their fingers and identify whether the graph is continuous (one smooth line) or discontinuous (broken into parts).

Assessment Ideas

Quick Check

After Station Rotation: The Function Zoo, present students with three graphs: one cubic, one reciprocal, and one exponential. Ask them to identify the function type for each, label any asymptotes or turning points, and explain their reasoning in writing.

Discussion Prompt

During Collaborative Investigation: The Growth Race, ask groups to share how they determined which function grows fastest over a given interval. Listen for clear explanations about the role of the gradient in exponential functions versus the constant increase in linear functions.

Exit Ticket

After Think-Pair-Share: The Asymptote Mystery, give each student a graph with a horizontal asymptote. Ask them to write the equation of the asymptote and explain why the graph cannot cross this line, using examples from their investigation.

Extensions & Scaffolding

• Challenge: Ask students to create a hybrid graph by combining a cubic and a reciprocal function, then predict and explain the behavior of the resulting graph.
• Scaffolding: Provide pre-labeled graphs with missing equations, and have students match each graph to its correct function by identifying key features like intercepts and asymptotes.
• Deeper exploration: Have students research and present on how cubic and exponential functions appear in fields like economics or biology, emphasizing their practical applications.

Key Vocabulary

Gradient (m)	A measure of the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
Y-intercept (c)	The point where a line crosses the y-axis. In the equation y=mx+c, it is represented by the value of c.
Slope-intercept form	A way of writing a linear equation as y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
Standard form	A way of writing a linear equation as ax + by = c, where 'a', 'b', and 'c' are constants.