Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Plot points on the Cartesian plane.
• Identify and interpret linear relationships from graphs.
• Represent simple relationships in tables and graphs.
• Determine input and output values from a graph.
• Identify independent and dependent variables.

Lesson summary

In Week 4, we delve deeper into the fascinating world of functions, graphs, and relationships. This isn't just abstract mathematics; it's a powerful tool for understanding and modelling patterns we see all around us, from the changing cellphone data costs over time to the relationship between rainfall and crop yield in our farming communities. Understanding functions and graphs allows us to predict trends, make informed decisions, and solve problems more effectively. Imagine you're trying to budget your airtime – a graph showing data usage versus cost can help you make the most of your money!

Lesson notes

The Cartesian Plane The Cartesian plane (also known as the coordinate plane) is formed by two number lines that intersect at right angles at a point called the origin. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. The x-axis represents the independent variable and the y-axis represents the dependent variable. Any point on the plane is identified by an ordered pair of numbers, written as (x, y). 'x' is the x-coordinate (also called the abscissa), and 'y' is the y-coordinate (also called the ordinate). The x-coordinate tells you how far to move horizontally from the origin, and the y-coordinate tells you how far to move vertically from the origin.

Remember: (x, y) – x comes first, then y!

The plane is divided into four quadrants: Quadrant I: x > 0, y > 0 (Both positive)

Quadrant II: x 0 (x negative, y positive)

Quadrant III: x 0, y < 0 (x positive, y negative)

Example 1: Plot the points A(2, 3), B(-1, 4), C(-3, -2), and D(4, -1) on the Cartesian plane.

Solution: A(2, 3): Start at the origin (0, 0). Move 2 units to the right along the x-axis, then 3 units up along the y-axis. B(-1, 4): Start at the origin. Move 1 unit to the left along the x-axis, then 4 units up along the y-axis. C(-3, -2): Start at the origin. Move 3 units to the left along the x-axis, then 2 units down along the y-axis. D(4, -1): Start at the origin. Move 4 units to the right along the x-axis, then 1 unit down along the y-axis. (Imagine or draw a Cartesian Plane with the points plotted) Functions and Relationships A relationship exists when two or more variables are connected. For example, the amount of electricity used in your house and the cost of your electricity bill have a relationship. A function is a special type of relationship where each input (x-value) has only one output (y-value). We often write this as y = f(x), where f(x) is the rule or formula that defines the function. Independent vs.

Dependent Variables: The independent variable (usually x) is the variable you control or choose. The dependent variable (usually y) is the variable that changes as a result of changes in the independent variable. In the electricity bill example, the amount of electricity used is the independent variable, and the cost is the dependent variable (because the cost depends on how much electricity you use). Representing Relationships with Tables and Graphs We can represent relationships in different ways: Table of Values: A table lists pairs of x and y values that satisfy the relationship.

Example: | x (Number of Airtime Vouchers) | y (Total Airtime Cost in Rands) | | ------------------------------- | -------------------------------- | | 1 | 30 | | 2 | 60 | | 3 | 90 | | 4 | 120 | In this table, x represents the number of airtime vouchers bought and y represents the total cost. Each voucher costs R

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0. Graph: A graph visually represents the relationship by plotting the (x, y) pairs from a table of values on the Cartesian plane and connecting them.

Example 2: Represent the airtime voucher cost relationship from the table above on a graph.

Solution: Choose your scale: The x-axis represents the number of vouchers (up to 4), and the y-axis represents the total cost (up to R120). Choose a scale that fits your data. For example, you could let each unit on the x-axis represent one voucher and each unit on the y-axis represent R

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0. Plot the points: Plot the points (1, 30), (2, 60), (3, 90), and (4, 120) on the Cartesian plane.

Connect the points: Since the relationship is linear (the cost increases at a constant rate), connect the points with a straight line. (Imagine or draw the graph, showing a straight line through the points)

Interpreting Graphs: Linear Relationships A linear relationship is represented by a straight line on a graph. The steepness of the line is called the slope or gradient.

Positive Slope: The line goes upwards from left to right (as x increases, y increases).

Negative Slope: The line goes downwards from left to right (as x increases, y decreases).

Zero Slope: The line is horizontal (y remains constant as x changes).

Undefined Slope: The line is vertical (x remains constant as y changes).

Example 3: Consider a graph showing the amount of water left in a tank as it drains over time. If the line slopes downwards from left to right, this indicates a negative slope. This means that as time (x) increases, the amount of water left in the tank (y) decreases. This illustrates a linear function. Guided Practice (With Solutions)

Question 1: Plot the following points on the Cartesian plane: P(-2, 1), Q(0, -3), and R(3, 0).

Solution: P(-2, 1): Start at the origin. Move 2 units left and 1 unit up. Q(0, -3): Start at the origin. Move 0 units left or right and 3 units down. This point lies on the y-axis. R(3, 0): Start at the origin. Move 3 units right and 0 units up or down. This point lies on the x-axis.