Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Define and identify functions in different forms.
• Use function notation, tables, and graphs.
• Identify independent and dependent variables.
• Draw graphs of linear functions (y = mx + c).
• Interpret information from simple graphs.

Lesson summary

Functions, graphs, and relationships are fundamental building blocks in mathematics. This topic is essential because it allows us to model and understand patterns and connections in the world around us. From predicting the spread of diseases to understanding financial growth and even planning optimal routes for taxi services, the principles of functions, graphs, and relationships are applied daily in South Africa. Imagine understanding how airtime costs increase with talk time, or how the price of mielie meal changes in response to weather patterns and demand! This understanding empowers you to make informed decisions and solve real-world problems.

Lesson notes

2.1 What is a Function? A function is a special type of relationship between two variables, often called input and output. For every input value, there is only one corresponding output value. Think of it like a vending machine. You put in a specific amount of money (the input), and you get out one specific snack or drink (the output). You wouldn't expect to put in R10 and get both a Coke and a packet of chips!

Input: The value we feed into the function (often denoted by x). This is also known as the independent variable.

Output: The value that comes out of the function (often denoted by y or f(x)). This is also known as the dependent variable, because its value depends on the input.

How to tell if something is a function: Tables: Look at the input values (usually the top row or first column). If any input value repeats with a different output value, it's NOT a function.

Graphs: Use the vertical line test. If any vertical line drawn on the graph intersects the graph more than once, it's NOT a function.

Equations: If you can solve the equation for y and for each x value you get only one y value, then it is a function. 2.2 Function Notation We often use function notation to represent functions concisely. Instead of writing "y = …", we write "f(x) = …". f(x) is read as "f of x." It means the value of the function f at the input value x. For example, if f(x) = 2x + 1, then f(3) means "the value of the function f when x is 3". To find f(3), we substitute 3 for x in the equation: f(3) = 2(3) + 1 = 6 + 1 = 7. 2.3 Representing Functions Functions can be represented in three main ways: Equations: A mathematical rule that describes the relationship between x and y.

Example: y = 3x - 2 or f(x) = 3x - 2 Tables of Values: A table that shows specific x values and their corresponding y values. | x | y | |---|---| | 0 | -2 | | 1 | 1 | | 2 | 4 | | 3 | 7 | Graphs: A visual representation of the function on a coordinate plane. The x-axis represents the input values, and the y-axis represents the output values. 2.4 Linear Functions: y = mx + c A linear function is a function whose graph is a straight line.

The general form of a linear function is: y = mx + c Where: m is the slope or gradient of the line. It represents how steep the line is. A positive m means the line slopes upwards from left to right, and a negative m means it slopes downwards. c is the y-intercept. It is the point where the line crosses the y-axis (i.e., the value of y when x is 0).

Example 1: The MTN airtime function Let's say MTN charges R1.50 per minute for calls.

This can be represented as a function: Cost (C) = R1.50 * Time (t) or C(t) = 1.5t If you talk for 5 minutes, the cost will be C(5) = 1.5 5 = R7.

5

0. The independent variable is time (t), and the dependent variable is cost (C).

Example 2: Filling a JoJo tank A JoJo tank starts with 100 litres of water. A tap fills it at a rate of 5 litres per minute. Let W be the amount of water in the tank (in litres) and t be the time (in minutes).

The function representing this is: W(t) = 5t + 100 After 10 minutes, the amount of water in the tank is: W(10) = 5(10) + 100 = 50 + 100 = 150 litres.

Example 3: The taxi fare equation A taxi charges a flat rate of R20 plus R5 per kilometre. Let F be the fare and d be the distance in kilometres.

The function representing this is: F(d) = 5d + 20 If you travel 12 km, the fare will be F(12) = 5(12) + 20 = 60 + 20 = R80 2.5 Drawing graphs of linear functions: Create a table of values: Choose a few x values (e.g., -2, -1, 0, 1, 2) and calculate the corresponding y values using the equation.

Plot the points: Plot the points (x, y) from your table on a coordinate plane.

Draw a line: Draw a straight line through the points. This line represents the graph of the function. Guided Practice (With Solutions)

Question 1: Is the following table of values a function? Explain your answer. | x | y | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | Solution: Yes, this is a function. Each x value has only one corresponding y value. There are no repeating x values with different y values.

Question 2: Given the function f(x) = 4x - 3, find f(2) and f(-1).

Solution: f(2) = 4(2) - 3 = 8 - 3 = 5 f(-1) = 4(-1) - 3 = -4 - 3 = -7 Therefore, f(2) = 5 and f(-1) = -

7. The key here is to substitute the given x value into the function's equation and then simplify.

Question 3: Draw the graph of the function y = 2x + 1 for x values between -2 and

2. Solution: First, create a table of values: | x | y = 2x + 1 | |---|---| | -2 | 2(-2) + 1 = -3 | | -1 | 2(-1) + 1 = -1 | | 0 | 2(0) + 1 = 1 | | 1 | 2(1) + 1 = 3 | | 2 | 2(2) + 1 = 5 | Now, plot the points (-2, -3), (-1, -1), (0, 1), (1, 3), and (2, 5) on a coordinate plane. Draw a straight line through these points. This line represents the graph of y = 2x +

1. Remember to label your axes.

Question 4: A cellphone company charges a fixed monthly fee of R50 plus R0.50 per megabyte (MB) of data used.