Graphing Functions: Linear and Quadratic

Active learning helps students grasp the dynamic nature of gradients and areas under curves. By manipulating graphs and discussing real-world contexts, students move beyond abstract rules to see how functions behave and change.

• Year 10
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Analyze how changing the slope and y-intercept of a linear function alters its graph's position and steepness.
2. Identify the roots (x-intercepts) and the turning point (vertex) of a quadratic function from its graph.
3. Compare the shapes and key features of linear and quadratic graphs, explaining their fundamental differences.
4. Calculate the coordinates of the turning point for a quadratic function given its equation.
5. Explain the graphical interpretation of the roots of a quadratic equation in the context of a real-world problem.

Inquiry Circle

⏱ 40 min·Small Groups

Inquiry Circle: The Racing Car

Groups use a motion sensor or a video of a car accelerating to create a distance-time graph. They must draw tangents at different points to estimate the speed at those moments and discuss how the speed is changing.

Prepare & details

Analyze how changes in the parameters of a linear function affect its graph.

Facilitation Tip: During the Racing Car investigation, circulate and ask groups to explain why the gradient changes as the curve steepens, pushing them to connect the visual with the mathematical idea.

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: Area Under the Curve

Students are given a velocity-time graph and asked to estimate the total distance using one large trapezium versus four smaller ones. They discuss in pairs why more trapeziums lead to a more accurate answer.

Prepare & details

Interpret the significance of the roots and turning point of a quadratic graph.

Facilitation Tip: For the Think-Pair-Share on area under the curve, assign each pair a different curve so the gallery of answers helps the whole class see the range of possible approaches.

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

⏱ 30 min·Pairs

Gallery Walk: Real-World Gradients

Stations feature different non-linear graphs (e.g., cooling of a cup of tea, bacterial growth). Students move in pairs to estimate the rate of change at specific points and explain what that rate means in the context of the problem.

Prepare & details

Compare the graphical properties of linear and quadratic functions.

Facilitation Tip: In the Gallery Walk, place graphs with clear real-world contexts at each station so students can focus on interpreting units and meanings rather than just calculating.

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

Teach gradients by having students physically draw tangents on printed curves, then measure and compare slopes. Emphasize that the tangent line is a local approximation, not the whole curve. For areas, start with simple shapes like rectangles and trapezoids before moving to curves, and always connect calculations back to what the area represents in context.

What to Expect

Students will confidently draw tangents to find gradients at points on curves and use the trapezium rule to approximate areas under curves. They will explain what these values represent in practical terms, such as acceleration or distance traveled.

Watch Out for These Misconceptions

Common Misconception

During the Collaborative Investigation: The Racing Car, watch for students drawing the same tangent line at every point on the curve.

What to Teach Instead

Ask students to place their rulers at a specific point on the curve and explain why the tangent line they drew only touches the curve at that one point. Have them compare slopes at different points to see the gradient changes.

Common Misconception

During the Gallery Walk: Real-World Gradients, watch for students ignoring the units on the axes when interpreting gradients or areas.

What to Teach Instead

Guide students to first identify the units on each axis, then ask them to write the units of the gradient (e.g., m/s²) and the area (e.g., m) before explaining what those units mean in context.

Assessment Ideas

Quick Check

After the Gallery Walk: Real-World Gradients, provide students with a mixed set of linear and quadratic graphs. Ask them to label the roots and turning point (if applicable) and write the equation for two linear examples. Collect this to check their ability to identify key features.

Exit Ticket

After the Think-Pair-Share: Area Under the Curve, give each student a card with a function (linear or quadratic). Ask them to sketch the graph, identify one key feature, and write one sentence explaining what that feature represents. Use these to assess their understanding of features and their real-world meaning.

Discussion Prompt

During the Collaborative Investigation: The Racing Car, pose the question: 'How does changing the 'c' value in y = mx + c affect the graph, and how does changing the 'a' value in y = ax² + bx + c affect its graph differently?' Facilitate a class discussion where students use their knowledge of graphs to explain the transformations.

Extensions & Scaffolding

• Challenge: Ask students to predict how the graph of y = x^2 changes if a constant is added inside the square, e.g., y = (x + 2)^2, and justify their prediction using the Racing Car context.
• Scaffolding: Provide a partially completed trapezium rule table with some values filled in, so students can focus on understanding the process rather than setting it up from scratch.
• Deeper exploration: Have students research how calculus is used in engineering or economics to design systems or analyze trends, then present their findings to the class.

Key Vocabulary

Linear Function	A function whose graph is a straight line. Its general form is y = mx + c, where m is the slope and c is the y-intercept.
Quadratic Function	A function of the form y = ax^2 + bx + c, where a is not zero. Its graph is a parabola.
Roots (x-intercepts)	The points where a graph crosses the x-axis. For a function, these are the values of x for which y = 0.
Turning Point (Vertex)	The highest or lowest point on a parabola. For y = ax^2 + bx + c, it indicates the maximum or minimum value of the function.
Slope (Gradient)	A measure of the steepness of a line. It is the ratio of the vertical change to the horizontal change between any two points on the line.
Y-intercept	The point where a graph crosses the y-axis. For a function, this is the value of y when x = 0.