Mathematics · Grade 10 · Quadratic Functions and Relations · Term 2

Introduction to Quadratic Functions

Students will define quadratic functions, identify their standard form, and recognize their parabolic graphs.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.HSF.IF.C.7.A

About This Topic

Properties of parabolas mark the transition from linear to quadratic thinking. Students learn to identify key features such as the vertex, axis of symmetry, direction of opening, and intercepts. This is a major component of the Ontario Grade 10 curriculum, as it introduces students to the concept of non-constant change and optimization. Understanding these properties is essential for modeling real world phenomena like the path of a ball or the shape of a satellite dish.

In a Canadian context, parabolas can be seen in the architecture of modern bridges or the trajectory of a puck in a hockey game. Discussing these familiar examples helps students see the practical utility of quadratic functions. This topic comes alive when students can physically model the patterns through graphing technology and collaborative investigations of real world shapes.

Key Questions

1. Differentiate between linear and quadratic functions based on their equations and graphs.
2. Explain why the graph of a quadratic function is always a parabola.
3. Analyze the significance of the 'a' coefficient in the standard form of a quadratic function.

Learning Objectives

• Identify the standard form of a quadratic function and classify functions as linear or quadratic based on their equations.
• Compare the graphical representations of linear and quadratic functions, recognizing the parabolic shape of quadratic graphs.
• Explain the relationship between the algebraic form of a quadratic function and the geometric properties of its parabolic graph.
• Analyze the effect of the coefficient 'a' on the width and direction of opening of a parabola.
• Differentiate between linear and quadratic functions by examining their equations and graphical characteristics.

Before You Start

Linear Functions and Their Graphs

Why: Students need a solid understanding of linear equations, slope, and graphing lines to effectively compare and contrast them with quadratic functions.

Basic Algebraic Manipulation

Why: Skills in simplifying expressions, substituting values, and solving simple equations are necessary for working with the standard form of quadratic functions.

Key Vocabulary

Quadratic Function	A function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. Its graph is a parabola.
Standard Form of a Quadratic Function	The form f(x) = ax^2 + bx + c, which clearly shows the coefficients a, b, and c.
Parabola	The U-shaped curve that is the graph of a quadratic function. It is symmetrical and opens either upwards or downwards.
Vertex	The highest or lowest point on a parabola, representing the maximum or minimum value of the function.
Axis of Symmetry	A vertical line that passes through the vertex of a parabola, dividing the parabola into two mirror-image halves.

Watch Out for These Misconceptions

Common Misconception

Confusing the vertex with the y intercept.

What to Teach Instead

Students often assume the 'starting point' of the graph is the vertex. Using a station rotation with different types of parabolas can help students see that the vertex is the turning point, which may or may not be on the y axis.

Common Misconception

Thinking that a larger 'a' value always means a 'wider' parabola.

What to Teach Instead

Actually, a larger 'a' value (like 5 vs 1) makes the parabola narrower. Collaborative graphing exercises help students see that a larger 'a' value causes the y values to increase faster, resulting in a steeper, narrower shape.

Active Learning Ideas

Gallery Walk: Parabola Scavenger Hunt

Students take photos of parabolic shapes around the school or in the community. They overlay a coordinate grid on their photos and identify the vertex, axis of symmetry, and intercepts of the 'real' parabola.

60 min·Small Groups

Think-Pair-Share: The 'a' Value Challenge

Pairs are given several quadratic equations with different 'a' values. They must predict how the 'a' value will change the shape and direction of the parabola before using graphing software to check their predictions.

30 min·Pairs

Inquiry Circle: Projectile Motion Simulation

Groups use a digital simulation to launch projectiles at different angles and speeds. They collect data on the path of the object and work together to identify the vertex (maximum height) and x intercepts (landing points).

50 min·Small Groups

Real-World Connections

• Engineers use quadratic functions to design the parabolic shape of satellite dishes and telescopes, which allows them to focus incoming signals or light to a single point.
• Athletes and coaches analyze the trajectory of projectiles, such as a basketball shot or a baseball hit, using quadratic functions to understand the path and optimize performance.
• Architects and bridge designers utilize the properties of parabolas when constructing suspension bridges, where the main cables often form a parabolic shape to distribute weight effectively.

Assessment Ideas

Quick Check

Provide students with a list of function equations. Ask them to identify which ones are quadratic and to write them in standard form if they are not already. Include a few linear functions for comparison.

Exit Ticket

On an index card, have students draw a simple sketch of a parabola opening upwards. Ask them to label the vertex and axis of symmetry. Then, have them write one sentence explaining how the 'a' coefficient would change if the parabola were narrower.

Discussion Prompt

Pose the question: 'How is the graph of a quadratic function different from the graph of a linear function?' Encourage students to discuss differences in shape, direction of change, and the number of x-intercepts.

Frequently Asked Questions

What is the vertex of a parabola?

The vertex is the highest or lowest point on a parabola, depending on whether it opens up or down. It is the 'turning point' where the graph changes direction. In real world terms, it often represents a maximum value (like peak height) or a minimum value (like lowest cost).

How can active learning help students understand parabolas?

Active learning, such as a 'Parabola Scavenger Hunt' or using motion sensors, allows students to connect the abstract properties of a quadratic equation to physical reality. When students see a parabola in a bridge or a fountain, identifying the vertex and axis of symmetry becomes a concrete task rather than a theoretical exercise, leading to deeper conceptual understanding.

What does the 'axis of symmetry' mean?

The axis of symmetry is an imaginary vertical line that passes through the vertex and divides the parabola into two perfectly mirrored halves. Every point on one side of the parabola has a corresponding point on the other side that is the same distance from this line.

Why do some parabolas open down?

A parabola opens down when the leading coefficient (the 'a' value in the equation) is negative. This reflection across the x axis happens because every positive y value from the parent function is multiplied by a negative number, flipping the entire shape upside down.