Mathematics · Secondary 3 · Functions and Graphs · Semester 1

Introduction to Functions

Defining functions, domain, range, and using function notation to represent relationships.

• MOE Syllabus Outcomes
• MOE: Numbers and Algebra - S3
• MOE: Functions and Graphs - S3

About This Topic

Quadratic functions introduce students to the world of non-linear relationships. In Secondary 3, the focus is on the parabola, its shape, symmetry, and key features like the turning point and intercepts. Students learn how the coefficients in the equation y = ax^2 + bx + c affect the graph: 'a' determines the width and direction (smiley or frowny face), 'c' is the y-intercept, and the relationship between 'a' and 'b' shifts the axis of symmetry.

This topic is highly visual and serves as a bridge to more advanced coordinate geometry. Understanding the 'turning point' as a maximum or minimum value has direct applications in optimization problems. This topic comes alive when students can use graphing software or physical drawing tools to see how changing a single number in an equation instantly transforms the curve. Structured peer explanation of these transformations helps solidify the link between algebra and geometry.

Key Questions

1. Explain the concept of a function as a special type of relation.
2. Differentiate between dependent and independent variables in a functional relationship.
3. Construct examples of real-world relationships that can be modeled as functions.

Learning Objectives

• Explain the definition of a function as a rule that assigns exactly one output to each input.
• Identify the domain and range of a function given a specific set of ordered pairs or a graph.
• Calculate the output value of a function for a given input value using function notation.
• Differentiate between a relation that is a function and one that is not, using the vertical line test.
• Construct a real-world scenario that can be represented by a function, defining its input and output variables.

Before You Start

Introduction to Relations and Ordered Pairs

Why: Students need to understand how to represent relationships using ordered pairs and coordinate points before they can classify them as functions.

Basic Algebraic Expressions

Why: Students must be able to substitute values into expressions and solve simple equations to work with function notation and evaluate functions.

Key Vocabulary

Function	A relation where each input value (from the domain) is associated with exactly one output value (in the range).
Domain	The set of all possible input values for which a function is defined.
Range	The set of all possible output values that a function can produce.
Function Notation	A way to represent a function using symbols, such as f(x), to denote the output of a function 'f' when the input is 'x'.
Independent Variable	The input variable in a function, typically represented by 'x', whose value does not depend on other variables in the function.
Dependent Variable	The output variable in a function, typically represented by 'y' or f(x), whose value depends on the independent variable.

Watch Out for These Misconceptions

Common Misconception

Thinking that a larger 'a' value makes the parabola wider.

What to Teach Instead

Actually, a larger 'a' makes the curve steeper and narrower. Using a 'stretching' analogy or having students plot points for y=x^2 and y=5x^2 side-by-side helps them see that the values grow much faster with a larger coefficient.

Common Misconception

Confusing the y-intercept with the turning point.

What to Teach Instead

Students often assume the lowest point of the curve must be on the y-axis. By looking at various graphs where the turning point is shifted left or right, students can visually distinguish between where the graph hits the axis (c) and where it turns.

Active Learning Ideas

Inquiry Circle: The Parabola Transformer

Using graphing software like Desmos, students work in pairs to explore how changing 'a', 'b', and 'c' shifts the parabola. They must complete a 'mission' to create a parabola that passes through specific points or has a specific turning point.

40 min·Pairs

Gallery Walk: Match the Graph to the Equation

Place several parabolas on the walls and give groups a set of equations. Students must use their knowledge of intercepts and turning points to match them, leaving a written justification for each match on the wall.

30 min·Small Groups

Think-Pair-Share: Symmetry Secrets

Show a graph with two x-intercepts. Ask students to find the x-coordinate of the turning point without the equation. After pairing, the class discusses how the axis of symmetry is always exactly halfway between the roots.

20 min·Pairs

Real-World Connections

• The cost of a taxi ride is a function of the distance traveled. The meter starts at a base fare (fixed input) and adds a charge for each kilometer or mile (independent variable), determining the total fare (dependent variable).
• The number of hours a student studies for an exam can be modeled as a function of their test score. While not perfectly linear, generally, more study hours (independent variable) lead to a higher score (dependent variable), up to a certain point.
• The amount of medicine dispensed by a machine can be a function of the number of times a button is pressed. Each press (input) results in a specific, consistent dose (output).

Assessment Ideas

Quick Check

Present students with a set of ordered pairs, e.g., {(1, 2), (2, 4), (3, 6), (1, 8)}. Ask: 'Is this set of pairs a function? Explain why or why not, referring to the definition of a function.'

Exit Ticket

Provide students with the function notation g(x) = 3x - 5. Ask them to: 1. Calculate g(4). 2. If g(a) = 10, what is the value of 'a'?

Discussion Prompt

Pose the scenario: 'A student is selling handmade bracelets for $5 each. What is the independent variable? What is the dependent variable? Write the relationship using function notation.'

Frequently Asked Questions

How can I tell if a parabola has a maximum or minimum point?

Look at the coefficient of the x-squared term (a). If 'a' is positive, the parabola opens upwards like a smile, meaning it has a minimum point at the bottom. If 'a' is negative, it opens downwards like a frown, resulting in a maximum point at the top.

What is the axis of symmetry in a quadratic function?

The axis of symmetry is a vertical line that passes through the turning point, dividing the parabola into two mirror-image halves. Its equation is always x = k, where k is the x-coordinate of the turning point.

How does active learning help students understand functions?

Functions can feel abstract when just looking at equations. Active learning, especially using dynamic graphing tools in a collaborative setting, allows students to see the 'cause and effect' of changing variables. This immediate visual feedback helps them build a mental map of how algebra controls geometry.

Why is the y-intercept always the 'c' value in y = ax^2 + bx + c?

The y-intercept occurs where x = 0. If you substitute x = 0 into the equation, both the ax^2 and bx terms become zero, leaving only 'c'. This is a quick and reliable way to find where any polynomial graph crosses the vertical axis.