Lesson Notes By Weeks and Term v5 - Grade 9

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 linear functions using tables, equations, and graphs.
• Determine the gradient and y-intercept of a linear function.
• Draw graphs of linear functions from their equations.
• Recognize and interpret simple non-linear graphs.
• Solve simple problems involving linear functions in real-life contexts.

Lesson summary

This week, we delve into the exciting world of functions and graphs! Understanding functions and graphs is crucial in mathematics because they help us model and understand relationships between different quantities. From calculating cell phone costs to predicting population growth, functions and graphs are powerful tools for problem-solving. In the South African context, these concepts are essential for understanding financial literacy, analysing data related to social and economic trends, and even making informed decisions about resource management.

Lesson notes

2.1 What is a Function?

A function is like a machine: you put something in (the input), and it gives you something else out (the output). Each input has only one output. We usually represent a function with a letter, like 'f', and write it as f(x). The 'x' is the input, and f(x) is the output. We can also think of a function as a relationship between an independent variable (usually x) and a dependent variable (usually y), where the value of y depends on the value of x. 2.2 Linear Functions A linear function is a function whose graph is a straight line.

The general form of a linear function 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 (how steep the line is) c is the y-intercept (where the line crosses the y-axis).

Understanding the Gradient (m): The gradient tells us how much y changes for every one unit change in x.

It can be calculated as: m = (change in y) / (change in x) = Δy / Δx = (y₂ - y₁) / (x₂ - x₁) where (x₁, y₁) and (x₂, y₂) are any two points on the line. If m is positive, the line slopes upwards from left to right. If m is negative, the line slopes downwards from left to right. If m = 0, the line is horizontal.

Understanding the y-intercept (c): The y-intercept is the point where the line crosses the y-axis. At this point, x =

0. So, the y-intercept is the value of y when x =

0. Example 1: Cellphone Data Costs A cellphone company charges a fixed monthly fee of R50 plus R2 per gigabyte (GB) of data used. Write an equation for the total monthly cost (y) in terms of the data used (x GB).

Solution: The fixed fee (R50) is the y-intercept (c). The cost per GB (R2) is the gradient (m).

Therefore, the equation is: y = 2x + 50 If you use 5 GB of data (x=5), the total cost would be: y = 2(5) + 50 = 10 + 50 = R60 Example 2: Finding the Gradient and y-intercept Consider the line passing through the points (1, 3) and (4, 9). Find the equation of this line.

Solution: Calculate the gradient (m): m = (y₂ - y₁) / (x₂ - x₁) = (9 - 3) / (4 - 1) = 6 / 3 = 2 Use the point-slope form of a linear equation: y - y₁ = m(x - x₁). We can use either point. Let's use (1, 3). y - 3 = 2(x - 1) Simplify to the slope-intercept form (y = mx + c): y - 3 = 2x - 2 y = 2x - 2 + 3 y = 2x + 1 Therefore, the equation of the line is y = 2x +

1. The gradient is 2, and the y-intercept is 1. 2.3 Simple Non-Linear Functions While linear functions produce straight lines, non-linear functions produce curves. We will briefly introduce a few simple non-linear functions. a)

Quadratic Functions (Parabolas): y = ax² + q The graph is a parabola (a U-shaped curve). If a is positive, the parabola opens upwards (minimum turning point). If a is negative, the parabola opens downwards (maximum turning point). q shifts the parabola up or down the y-axis.

Example: y = x² (a simple parabola opening upwards). b)

Hyperbolic Functions: y = a/x + q The graph is a hyperbola, consisting of two separate curves called branches. The lines x=0 and y=q are asymptotes (the graph gets closer and closer to these lines but never touches them).

Example: y = 1/x (a simple hyperbola). c)

Exponential Functions: y = aˣ The graph shows rapid growth or decay. The graph always passes through the point (0,1) if the function is simply y = aˣ The x-axis is an asymptote.

Example: y = 2ˣ (an exponential growth function). Important

Note: We are introducing these non-linear functions here. A deeper dive will come in later grades. The focus this week is primarily on recognizing their shapes and understanding that they are not linear. Guided Practice (With Solutions)

Question 1: A taxi charges a fixed call-out fee of R25 plus R8 per kilometer travelled. a) Write an equation for the total cost (y) in terms of the distance travelled (x km). b) Calculate the total cost for a journey of 10 km. c) If a passenger pays R81, how far did they travel?

Solution: a) y = 8x + 25 (The gradient is R8 per km, and the y-intercept is R25) b) y = 8(10) + 25 = 80 + 25 = R105 c) 81 = 8x + 25 81 - 25 = 8x 56 = 8x x = 56/8 = 7 km Question 2: The graph of a linear function passes through the points (0, -2) and (2, 2). a) Find the gradient of the line. b) Write the equation of the line in the form y = mx + c.

Solution: a) m = (2 - (-2)) / (2 - 0) = 4 / 2 = 2 b) Since the line passes through (0, -2), the y-intercept c is -

2. Therefore, the equation is y = 2x -

2. Question 3: Identify which of the following equations represent linear functions: a) y = 3x - 5 b) y = x² + 2 c) y = 4/x d) y = -2x + 1 Solution: a) Linear (in the form y = mx + c) b) Non-Linear (quadratic, x²) c) Non-Linear (hyperbolic, x in the denominator) d) Linear (in the form y = mx + c) Independent Practice (Questions Only) A market vendor sells mangoes for R5 each. Write an equation that relates the total cost (y) to the number of mangoes purchased (x). Find the gradient and y-intercept of the line y = -3x + 7.