Mathematics · Year 9 · Linear and Non Linear Relationships · Term 2

Graphing Linear Functions (y=mx+c)

Students will sketch linear functions using the gradient-intercept form, identifying the y-intercept and gradient.

• ACARA Content Descriptions
• AC9M9A05

About This Topic

The introduction to parabolas marks a significant shift from linear to non-linear thinking in Year 9. Students explore the unique symmetrical 'U' shape of quadratic relationships and how they differ from the constant rate of change seen in straight lines. This topic is essential for understanding physics (projectile motion), architecture (archways), and even economics (profit curves). By investigating the basic properties of y = x^2, students begin to appreciate the complexity and beauty of non-linear functions.

In the Australian Curriculum, this unit encourages students to use technology to investigate how changing coefficients affects the graph's shape. This topic is particularly well-suited to active learning, where students can use motion sensors to 'walk' a parabola or use digital graphing tools to collaboratively explore transformations. Seeing these curves in the real world, from the Sydney Harbour Bridge to a bouncing ball, helps ground the abstract algebra in reality.

Key Questions

1. Analyze how changing the 'm' and 'c' values in y=mx+c affects the graph of a line.
2. Construct a linear equation from a given graph.
3. Justify why the y-intercept is a crucial point for sketching linear graphs.

Learning Objectives

• Identify the y-intercept and gradient from a linear equation in the form y=mx+c.
• Sketch the graph of a linear function by plotting the y-intercept and using the gradient to find a second point.
• Analyze how changes in the gradient ('m') and y-intercept ('c') affect the position and steepness of a linear graph.
• Construct a linear equation in the form y=mx+c given the graph of a straight line.
• Justify the significance of the y-intercept as a starting point for graphing linear functions.

Before You Start

Plotting Points and Coordinate Grids

Why: Students must be able to accurately plot points on a Cartesian plane to sketch graphs.

Understanding Variables and Expressions

Why: Students need to understand the concept of variables (x and y) and how they relate in an algebraic expression to work with linear equations.

Key Vocabulary

Gradient (m)	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. It indicates the rate of change.
Y-intercept (c)	The point where a line crosses the y-axis. In the equation y=mx+c, 'c' represents the y-coordinate of this point, often written as (0, c).
Linear function	A function whose graph is a straight line. It can be represented by an equation in the form y=mx+c, where m and c are constants.
Gradient-intercept form	The standard form of a linear equation, y=mx+c, where 'm' is the gradient and 'c' is the y-intercept.

Watch Out for These Misconceptions

Common Misconception

Students often think the bottom of a parabola is a sharp 'V' shape like an absolute value graph.

What to Teach Instead

This happens because they only plot a few points. Encouraging them to plot points closer to the origin (like 0.5 and -0.5) or using digital tools helps them see the smooth, curved nature of the turning point. Peer-sharing of graphs helps highlight this difference.

Common Misconception

Believing that a negative 'a' value means the whole graph is in the negative y-region.

What to Teach Instead

Students may think the graph just 'moves down'. By calculating values for y = -x^2, they see that the graph 'flips' or reflects across the x-axis. Active exploration with graphing software is the fastest way to correct this visual misunderstanding.

Active Learning Ideas

Simulation Game: Projectile Motion Capture

Students film a peer throwing a basketball in an arc. Using slow-motion playback or graphing software, they plot the path of the ball on a coordinate grid to see the parabolic shape. They then discuss why the path isn't a straight line, linking it to gravity.

50 min·Small Groups

Inquiry Circle: The 'a' Value Exploration

Using digital graphing software, groups are assigned different values for 'a' in y = ax^2. They must observe and record what happens as 'a' gets larger, smaller, or negative. They then present their findings to create a class 'rule' for transformations.

35 min·Small Groups

Gallery Walk: Parabolas in the Wild

Students bring in photos of parabolic shapes they've found in nature or architecture (fountains, bridges, satellite dishes). They display these and explain where the 'turning point' and 'axis of symmetry' would be on their photo. This builds visual recognition of quadratic forms.

30 min·Individual

Real-World Connections

• City planners use linear equations to model population growth or traffic flow over time, helping to predict future needs for infrastructure and services.
• Economists use linear functions to represent simple cost or revenue models, such as the cost of producing a certain number of items or the revenue generated from selling them at a fixed price per unit.

Assessment Ideas

Quick Check

Provide students with 3-4 linear equations (e.g., y=2x+1, y=-x+3, y=0.5x-2). Ask them to identify the gradient and y-intercept for each and sketch the corresponding graph on mini-whiteboards. Review sketches for accuracy of intercept placement and direction of gradient.

Exit Ticket

Give students a graph of a straight line that passes through (0, -2) and (4, 4). Ask them to write the linear equation for this line in y=mx+c form and explain in one sentence why the y-intercept is a useful starting point for sketching.

Discussion Prompt

Pose the question: 'If you have two linear equations, y=3x+5 and y=3x-2, what do their graphs have in common, and how do they differ? What does this tell you about the role of 'c' in the equation?' Facilitate a class discussion focusing on parallel lines and the meaning of the y-intercept.

Frequently Asked Questions

What makes a relationship 'quadratic' rather than 'linear'?

In a linear relationship, the rate of change is constant. In a quadratic relationship, the rate of change is itself changing. Visually, this means a parabola gets steeper and steeper as you move away from the turning point, whereas a line stays at the same angle.

Where do we see parabolas in everyday life in Australia?

The most famous example is the Sydney Harbour Bridge (though it's technically a catenary, it's often modeled as a parabola in school maths). You also see them in the path of a kicked footy, the shape of a satellite dish, and even in the arc of water from a garden sprinkler.

Why is the turning point so important?

The turning point (or vertex) represents the maximum or minimum value of the function. In real-world terms, this could be the highest point a ball reaches or the point where a business makes the least amount of profit before things improve.

How can active learning help students understand parabolas?

Active learning, such as capturing projectile motion or using interactive graphing software, allows students to see the immediate impact of changing variables. Instead of just plotting points on paper, they can 'play' with the curve. This experimentation builds a much stronger mental model of how quadratic functions behave, making the transition to more complex quadratic algebra in Year 10 much smoother.