Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Represent relationships using tables, flow diagrams, and equations.
• Draw and interpret straight-line graphs.
• Understand the properties of linear functions (y = mx + c).
• Solve real-world problems involving linear functions.

Lesson summary

This week, we delve into the fascinating world of functions, graphs, and the relationships they represent. Understanding functions and graphs is crucial because they are the visual language of mathematics and are used to model real-world situations, predict trends, and solve problems across various fields, from science and economics to technology and everyday budgeting. Imagine tracking your mobile data usage over time – that's a function displayed on a graph! Or predicting the rainfall patterns for the upcoming planting season based on historical data – that also involves understanding functional relationships.

Lesson notes

2.1 What is a Function? A function is a special kind of relationship between two sets of elements (usually numbers).

Think of it like a machine: you put something in (the input, also known as the independent variable), and the machine does something to it and gives you something out (the output, also known as the dependent variable). For it to be a true function, each input can only have one output. We often call the input 'x' and the output 'y'. The function itself defines the rule that connects 'x' and 'y'. 2.2 Representing Functions We can represent functions in several ways: Tables: A table shows pairs of 'x' and 'y' values that satisfy the function's rule.

Example: Consider the function y = 2x +

1. A table representing this function could be: | x | y | |---|---| | -2 | -3 | | -1 | -1 | | 0 | 1 | | 1 | 3 | | 2 | 5 | Flow Diagrams: A flow diagram shows the steps involved in transforming the input (x) into the output (y).

Example: For the function y = 2x + 1, the flow diagram would be: Input (x) --> Multiply by 2 --> Add 1 --> Output (y)

Algebraic Equations: This is the most concise way to represent a function. It directly shows the relationship between 'x' and 'y'.

Example: y = 2x + 1 2.3 Linear Functions and Straight-Line Graphs A linear function is a function whose graph is a straight line.

The general form of a linear equation is: y = mx + c Where: y is the dependent variable (output) x is the independent variable (input) m is the gradient (or slope) of the line. It tells us how steep the line is and whether it's increasing or decreasing. A positive gradient means the line goes up from left to right, while a negative gradient means it goes down. The gradient can be calculated as change in y / change in x. c is the y-intercept. It's the point where the line crosses the y-axis (the vertical axis). The coordinates of the y-intercept are always (0, c). 2.4 Understanding Gradient (m) The gradient 'm' is crucial. It represents the rate of change. For every 1 unit increase in 'x', 'y' changes by 'm' units.

Example: If m = 3, for every 1 you increase x by, y increases by 3. 2.5 Understanding y-intercept (c) The y-intercept 'c' is simply the value of 'y' when 'x' is zero. It's the starting point of the line on the y-axis. 2.6 Plotting a Straight-Line Graph To draw a straight-line graph, you need at least two points.

You can find these points by: Choosing values for x: Pick any two values for 'x' (e.g., x = 0 and x = 1).

Calculating the corresponding y values: Substitute the chosen 'x' values into the equation to find the corresponding 'y' values.

Plotting the points: Plot the (x, y) coordinates on a graph.

Drawing the line: Use a ruler to draw a straight line through the two points.

Worked example

Example 1: Given the equation, find points and plot the graph.

Equation: y = x + 2

Let x = 0: y = 0 + 2 =

2. Point: (0, 2)

Let x = 1: y = 1 + 2 =

3. Point: (1, 3)

Plot the points (0, 2) and (1, 3) on a graph and draw a straight line through them.

The gradient 'm' is 1 (because y = 1x + 2) and the y-intercept 'c' is

2. Example 2: Given points, find the equation.

Points: (1, 4) and (2, 7)

Calculate the gradient: m = (change in y) / (change in x) = (7 - 4) / (2 - 1) = 3/1 = 3

We know the equation is y = 3x + c. Substitute one of the points to find 'c'. Let's use (1, 4): 4 = 3(1) + c.

Therefore, c = 4 - 3 = 1

The equation is y = 3x + 1