Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent relationships between two variables using tables.
• Draw and interpret line graphs for real-life situations.
• Identify dependent and independent variables.
• Use a function to find output values from given input values.

Lesson summary

Functions, graphs, and relationships are fundamental building blocks in mathematics. Understanding them allows us to describe and predict patterns in the world around us. In South Africa, this is particularly relevant for analysing trends in population growth, economic indicators, and even sports statistics like cricket scores or soccer league tables. Being able to interpret and create graphs helps us make informed decisions and understand complex information presented in visual formats by the media and government. This week, we will focus on recognizing and representing relationships between variables, interpreting graphs, and using tables to represent functions.

Lesson notes

2.1 Variables: A variable is a symbol (usually a letter) that represents a quantity that can change or vary. There are two main types of variables in a relationship: Independent Variable: This is the variable that we can control or choose. It is often called the input variable.

Dependent Variable: This is the variable that changes because of the independent variable. Its value depends on the value of the independent variable. It is often called the output variable.

Example: Consider the relationship between the number of hours you work at a part-time job and the amount of money you earn. The independent variable is the number of hours you work (you choose how many hours). The dependent variable is the amount of money you earn (how much you earn depends on how many hours you worked). 2.2 Functions: A function is a rule or a relationship that assigns to each input value (from the independent variable) exactly one output value (from the dependent variable). We often write functions using equations.

Example: The function y = 2x + 1 represents a relationship where y (the dependent variable) is calculated by multiplying x (the independent variable) by 2 and then adding 1. 2.3 Representing Relationships with Tables: A table is a useful way to show the relationship between two variables. The independent variable is usually listed in the first row or column, and the corresponding dependent variable is listed in the second row or column.

Example: Let's use the function y = 2x + 1 to create a table. | x (Input) | y (Output) | |---|---| | 0 | 1 | | 1 | 3 | | 2 | 5 | | 3 | 7 | To generate these values, we substitute the values of x into the equation y = 2x + 1: When x = 0, y = (2 0) + 1 = 1 When x = 1, y = (2 1) + 1 = 3 When x = 2, y = (2 2) + 1 = 5 When x = 3, y = (2 3) + 1 = 7 2.4 Representing Relationships with Line Graphs: A line graph is a visual way to represent the relationship between two variables. The independent variable is usually plotted on the horizontal axis (x-axis), and the dependent variable is plotted on the vertical axis (y-axis). Each pair of values from the table is plotted as a point on the graph. The points are then connected with a line.

Example: Let's graph the function y = 2x + 1 using the table we created earlier. Draw the x and y axes. Label the x-axis "Input (x)" and the y-axis "Output (y)". Choose appropriate scales for both axes. In this case, we can use intervals of 1 for both axes. Plot the points (0, 1), (1, 3), (2, 5), and (3, 7). Draw a straight line through the points. The resulting line graph shows the linear relationship between x and y. 2.5 Interpreting Graphs: Interpreting a graph involves understanding the information it presents. We can use graphs to answer questions about the relationship between the variables.

Example: A graph shows the temperature of water as it is heated over time. What was the initial temperature of the water? Look at the y-axis value when the x-axis value (time) is

0. How long did it take for the water to reach boiling point (100°C)? Find 100°C on the y-axis and trace it across to the line. Then, read the corresponding value on the x-axis (time). What was the temperature of the water after 5 minutes? Find 5 minutes on the x-axis and trace it up to the line. Then, read the corresponding value on the y-axis (temperature). Guided Practice (With Solutions)

Question 1: A taxi charges a fixed call-out fee of R15 plus R8 per kilometre travelled. Create a table showing the cost for distances of 0km, 1km, 2km, and 3km.

Solution: Let x be the distance travelled (in km). Let y be the total cost (in Rand). The function representing this relationship is y = 8x + 15. | x (km) | y (Rand) | |---|---| | 0 | 15 | | 1 | 23 | | 2 | 31 | | 3 | 39 | Explanation: We substitute each value of x into the equation y = 8x + 15 to find the corresponding value of y. For example, when x = 2, y = (8 2) + 15 = 16 + 15 =

3

1. Question 2: Identify the independent and dependent variables in the following scenario: The number of hours studied and the grade achieved on a test.

Solution: Independent Variable: Number of hours studied. (You choose how long you study).

Dependent Variable: Grade achieved on the test. (Your grade depends on how much you studied).

Explanation: The grade you achieve on the test depends on the number of hours you study.

Question 3: The graph below shows the distance a delivery biker travels over time. (Assume a simple graph with Time (minutes) on the x-axis and Distance (km) on the y-axis.

Points: (0,0), (10, 5), (20, 10), (30, 15)) a) What distance had the biker travelled after 20 minutes? b) How long did it take the biker to travel 15 km?

Solution: a) After 20 minutes, the biker had travelled 10 km. b) It took the biker 30 minutes to travel 15 km.

Explanation: To answer these questions, we read the values directly from the graph. For part (a), we find 20 minutes on the x-axis, go up to the line, and then read the corresponding distance on the y-axis.