Mathematics · Secondary 3 · Functions and Graphs · Semester 1

Linear Functions and Graphs Review

Reviewing gradient, y-intercept, and different forms of linear equations (y=mx+c, ax+by=c).

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

About This Topic

Beyond quadratics, Secondary 3 students explore power functions (like cubic and reciprocal graphs) and simple exponential functions. These graphs introduce new behaviors, such as the 'S-shape' of a cubic or the 'asymptotes' of a reciprocal function where the graph approaches a line but never touches it. Exponential functions, in particular, are vital for understanding real-world phenomena like compound interest, population growth, and the spread of viruses.

In the MOE syllabus, the focus is on recognizing the general shape of these functions and identifying their key features. Students learn to distinguish between growth and decay in exponential models. This topic is best taught through comparative investigations where students plot different types of functions on the same axes to see how they grow at different rates. Students grasp these concepts faster through peer explanation and by categorizing different 'families' of graphs based on their equations.

Key Questions

1. Analyze how the gradient and y-intercept define a linear function's graph.
2. Compare the advantages of using different forms of linear equations.
3. Predict the behavior of a linear graph given changes in its gradient or y-intercept.

Learning Objectives

• Analyze how changes in the gradient and y-intercept of a linear function affect the position and steepness of its graph.
• 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.
• Calculate the gradient and y-intercept for a given linear equation in any form.
• Explain the relationship between the algebraic representation of a linear function and its graphical representation.
• Predict the intersection point of two linear graphs given their equations.

Before You Start

Introduction to Coordinate Geometry

Why: Students need to be familiar with plotting points and understanding the Cartesian plane to visualize linear graphs.

Solving Linear Equations

Why: Students must be able to manipulate and solve single-variable linear equations to isolate variables and find unknown values.

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.

Watch Out for These Misconceptions

Common Misconception

Thinking that an exponential graph will eventually touch the x-axis.

What to Teach Instead

Students often draw the tail of an exponential decay curve touching zero. Using a calculator to show that even a very small number like 0.5 to the power of 100 is still not zero helps them understand the concept of an asymptote.

Common Misconception

Confusing the shapes of cubic and reciprocal graphs.

What to Teach Instead

Students may mix up the 'S' curve with the two separate branches of a reciprocal. Having students physically trace the curves and identify if they are 'continuous' (one smooth line) or 'discontinuous' (broken into parts) helps distinguish them.

Active Learning Ideas

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.

50 min·Small Groups

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.

35 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.

20 min·Pairs

Real-World Connections

• City planners use linear functions to model the relationship between the number of new housing units built and the required increase in public transportation routes. The gradient indicates how many new bus lines are needed per 100 houses, and the y-intercept might represent existing infrastructure.
• Economists analyze supply and demand curves, which are often linear, to predict market equilibrium points. The gradient of the supply curve shows how much producers are willing to supply at different prices, while the demand curve gradient shows how much consumers will buy.

Assessment Ideas

Quick Check

Present students with three linear equations: y = 2x + 5, 3x + y = 1, and y - 1 = 4(x - 2). Ask them to rewrite the second and third equations in slope-intercept form and identify the gradient and y-intercept for all three.

Discussion Prompt

Pose the question: 'Imagine you are designing a simple video game where a character moves in a straight line. How would you use the gradient and y-intercept to control the character's speed, direction, and starting position on the screen?' Facilitate a class discussion where students share their ideas.

Exit Ticket

Give each student a graph of a line. Ask them to write the equation of the line in both slope-intercept form and standard form. Then, ask them to describe in one sentence how doubling the gradient would change the graph.

Frequently Asked Questions

What is an asymptote in a reciprocal graph?

An asymptote is a line that a curve approaches but never actually reaches or crosses. For the graph of y = 1/x, the x-axis and y-axis are asymptotes because x can never be zero (you can't divide by zero) and y can never be zero (1 divided by anything is never zero).

How do exponential graphs differ from power graphs?

In a power graph (like y = x^2), the base is the variable and the exponent is a constant. In an exponential graph (like y = 2^x), the base is a constant and the variable is in the exponent. This small change leads to much faster growth in exponential functions.

How can active learning help students understand complex graphs?

Active learning, such as the 'Growth Race' investigation, allows students to see the behavior of functions through data they generate themselves. Instead of just looking at a finished curve, they experience the 'explosion' of exponential growth, which makes the steepness of the graph more meaningful.

Why does a cubic graph have an 'S' shape?

A cubic graph like y = x^3 has an 'S' shape because when you cube a negative number, it stays negative, and when you cube a positive number, it stays positive. This creates two arms pointing in opposite directions, with a 'twist' or inflection point in the middle.