Graphing Quadratic Functions

Graphing quadratic functions is a visual and tactile topic where students benefit from building curves themselves rather than memorizing steps. Active learning works here because plotting points in order, comparing shapes, and discussing shifts deepen understanding of symmetry and transformations more than passive reading or listening.

• Secondary 3
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Calculate the coordinates of the vertex of a quadratic function using the formula x = -b/(2a).
2. Plot the graph of a quadratic function by selecting appropriate x-values for a table of values.
3. Identify the y-intercept, x-intercepts, and axis of symmetry from the graph of a quadratic function.
4. Predict the effect of changing the constant term in a quadratic equation on the position of its graph.
5. Compare the shapes of different parabolas and explain how the coefficient 'a' influences the width and direction of opening.

Collaborative Problem-Solving

⏱ 25 min·Pairs

Pairs Plotting: Vertex First Challenge

Pairs select a quadratic equation and calculate the vertex coordinates first. They build a table with four x-values symmetric around the vertex, plot points on graph paper, and sketch the parabola. Partners check each other's axis of symmetry and discuss any asymmetries.

Prepare & details

Design a table of values that effectively captures the key features of a parabola.

Facilitation Tip: During Pairs Plotting, circulate and ask each pair to explain why they chose their x-values around the vertex before they plot.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Small Groups: Transformation Match-Up

Provide cards with quadratic equations and their graphs. Groups match them by predicting shifts from changes in coefficients, then verify by plotting one example per equation on shared paper. Discuss why certain changes stretch or reflect the parabola.

Prepare & details

Justify the importance of plotting the turning point accurately.

Facilitation Tip: For Transformation Match-Up, provide colored pencils so groups can trace and compare graphs side-by-side.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 40 min·Whole Class

Whole Class: Prediction Parade

Display an equation on the board. Students predict and sketch the graph individually on slips of paper, then parade to tape sketches under the correct description (e.g., 'vertex at (2, -1)'). Reveal the accurate plot and tally predictions.

Prepare & details

Predict how changes in the constant term affect the position of the parabola.

Facilitation Tip: In Prediction Parade, pause after each student shares their guess to ask the class to justify or challenge the prediction using the equation.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 20 min·Individual

Individual: Table Tweaks

Give students a poorly designed table for a quadratic. They revise it to include the vertex and symmetric points, replot the graph, and note improvements in accuracy. Submit before-and-after graphs with justifications.

Prepare & details

Design a table of values that effectively captures the key features of a parabola.

Facilitation Tip: During Table Tweaks, have students swap their tables with a partner to verify the symmetry before plotting.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Start with concrete examples before abstract formulas. Have students graph y = x^2 by hand first, then y = -x^2, so they see the effect of ‘a’ immediately. Avoid teaching vertex form too early; let students discover the vertex formula through repeated calculations in tables. Research shows that students retain transformations better when they manipulate graphs physically, so provide grid paper and rulers for accuracy.

What to Expect

By the end of these activities, students should confidently identify the vertex, axis of symmetry, and intercepts from any quadratic equation. They should also explain how changes to coefficients affect the graph’s position and shape, using precise vocabulary like ‘vertical shift’ and ‘direction of opening’ in their discussions.

Watch Out for These Misconceptions

Common Misconception

During Pairs Plotting, watch for students who assume all parabolas open upwards.

What to Teach Instead

Have these pairs graph y = x^2 and y = -x^2 side-by-side, then ask them to compare the two curves and explain how the coefficient a determines the direction of opening based on their observations.

Common Misconception

During Pairs Plotting, watch for students who place the vertex at x=0 without calculation.

What to Teach Instead

Prompt them to calculate the vertex using -b/(2a) for their given equation, then justify the x-coordinate’s position by comparing it to the linear term in the equation.

Common Misconception

During Transformation Match-Up, watch for students who think changing the constant term shifts the graph horizontally.

What to Teach Instead

Have them plot y = x^2 + 2 and y = (x + 2)^2 next to each other, then ask which graph moved vertically and which moved horizontally, using their plotted points to correct the misunderstanding.

Assessment Ideas

Quick Check

After Pairs Plotting, provide the equation y = 3x^2 + 6x - 2 and ask students to calculate the vertex’s x-coordinate, identify the y-intercept, and sketch the axis of symmetry on a mini whiteboard. Collect these to assess accuracy and reasoning.

Exit Ticket

After Transformation Match-Up, give students the equations y = x^2 - 4 and y = (x - 2)^2 + 1. Ask them to describe in one sentence how the two graphs differ in position and to identify the vertex for each.

Discussion Prompt

During Prediction Parade, pose the question: 'Why does including the vertex in your table of values help you graph a quadratic function accurately?' Facilitate a 3-minute class discussion, listening for references to symmetry and turning points in their explanations.

Extensions & Scaffolding

• Challenge students to write a new quadratic equation that shifts their plotted parabola 4 units up and 2 units left, then graph it without using a table.
• For struggling students, provide pre-labeled axes and partially completed tables with 3-4 points filled in to focus on symmetry.
• Deeper exploration: Ask students to graph two quadratics with the same vertex but different directions of opening, then compare their axes of symmetry and discuss why they align.

Key Vocabulary

Parabola	The U-shaped curve that is the graph of a quadratic function. It is symmetrical about a vertical line.
Vertex	The turning point of a parabola, which is either the minimum or maximum point on the graph.
Axis of Symmetry	The vertical line that divides the parabola into two mirror-image halves. It passes through the vertex.
Y-intercept	The point where the graph crosses the y-axis. For a quadratic function, this occurs when x = 0.
X-intercepts	The points where the graph crosses the x-axis. These are also known as the roots or zeros of the quadratic function.