Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Represent linear functions as equations, tables, and graphs.
• Identify the slope and y-intercept of a line.
• Sketch graphs of simple non-linear functions like y = x² and y = 1/x.
• Find the equation of a line from its graph or points.
• Solve real-world problems using linear functions.

Lesson summary

This week, we delve into the fascinating world of functions and graphs, specifically focusing on linear and simple non-linear functions. Understanding functions and graphs is fundamental to mathematics and has wide-ranging applications in everyday life. From predicting cellphone data usage costs based on consumption to understanding the trajectory of a cricket ball, functions and graphs are powerful tools for modelling and interpreting real-world scenarios. In South Africa, these concepts are used extensively in fields like economics (analysing growth trends), science (modelling weather patterns), and engineering (designing infrastructure).

Lesson notes

2.1 What is a Function? A function is a relationship between two sets of numbers (usually x and y) where each input (x-value) has exactly one output (y-value).

We can think of it as a machine: you put something in (x), and the machine performs an operation to give you something else out (y). We often write this as y = f(x), where f is the function.

Independent Variable (x): The input value. Also known as the domain.

Dependent Variable (y): The output value, which depends on the input. Also known as the range. 2.2 Linear Functions A linear function is a function that forms a straight line when graphed.

It can be represented by the equation: y = mx + c Where: m is the slope (gradient) of the line – how steep it is. It's the change in y divided by the change in x (rise over run). m can be positive (line goes up from left to right), negative (line goes down from left to right), zero (horizontal line), or undefined (vertical line). c is the y-intercept – the point where the line crosses the y-axis (when x = 0).

Example 1: Understanding Slope Imagine you are hiking up Table Mountain. The slope represents how steep the path is. A steeper path means a larger slope value. Let's say you walk 10 meters horizontally (change in x = 10m) and climb 2 meters vertically (change in y = 2m). The slope of the path is m = 2/10 = 0.

2. Example 2: Finding the Equation of a Linear Function Suppose a taxi charges a fixed call-out fee of R20 and R10 per kilometre travelled. This is a linear relationship. The fixed fee (R20) is the y-intercept (c = 20). The cost per kilometre (R10) is the slope (m = 10).

Therefore, the equation representing the taxi fare (y) for x kilometres travelled is: y = 10x +

2

0. If you travel 5km, the fare is y = 10(5) + 20 = R70. 2.3 Non-Linear Functions: Simple Quadratic Function (y = x²) A non-linear function does not form a straight line when graphed. A simple example is the quadratic function y = x². Its graph is a U-shaped curve called a parabola. Key Features of y = x²: Vertex: The lowest point of the parabola (in this case, (0, 0)).

Symmetry: The graph is symmetrical about the y-axis.

Values: y is always positive or zero, regardless of the value of x.

Example 3: Graphing y = x² Let's create a table of values and then plot the points: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | |------|----|----|----|---|---|---|---| | y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 | Plotting these points will reveal the parabolic shape. 2.4 Non-Linear Functions: Simple Reciprocal Function (y = 1/x) Another simple non-linear function is the reciprocal function y = 1/x. Key Features of y = 1/x: Asymptotes: The graph has asymptotes at x = 0 and y =

0. The graph gets infinitely close to these lines but never touches them.

Values: y is never zero. When x is a large positive number, y is a small positive number. When x is a large negative number, y is a small negative number.

Quadrants: The graph exists in the first and third quadrants.

Example 4: Graphing y = 1/x Let's create a table of values and then plot the points: | x | -3 | -2 | -1 | -0.5 | 0.5 | 1 | 2 | 3 | |------|------|------|------|------|------|-----|-----|-----| | y = 1/x | -0.33 | -0.5 | -1 | -2 | 2 | 1 | 0.5 | 0.33| Plotting these points will reveal the characteristic shape of a reciprocal function. Notice how the y values become very large or very small as x gets closer to

0. Guided Practice (With Solutions)

Question 1: A cellphone company charges R1.50 per minute for calls. Represent this relationship as an equation, a table of values (for 0, 1, 2, and 3 minutes), and a graph.

Solution: Equation: y = 1.5x (where y is the total cost and x is the number of minutes).

Table of Values: | Minutes (x) | Cost (y) | |-------------|----------| | 0 | 0 | | 1 | 1.50 | | 2 | 3.00 | | 3 | 4.50 | Graph: Plot the points from the table on a graph with minutes on the x-axis and cost on the y-axis. Draw a straight line through these points.

Commentary: This question introduces the concept of a linear function in a relatable context. The equation is straightforward, and the table of values helps visualise the relationship. The graph provides a visual representation of the linear function.

Question 2: A line passes through the points (1, 3) and (2, 5). Find the equation of the line.

Solution: Calculate the slope (m): m = (y₂ - y₁) / (x₂ - x₁) = (5 - 3) / (2 - 1) = 2/1 = 2 Use the slope-intercept form (y = mx + c) and one of the points to find the y-intercept (c): Let's use the point (1, 3). 3 = 2(1) + c 3 = 2 + c c = 1 Write the equation: y = 2x + 1

Commentary: This question requires applying the formula for slope and then using the slope-intercept form to determine the equation of the line. It reinforces the understanding of m and c.

Question 3: Sketch the graph of y = x² -

1. Describe how it differs from the graph of y = x².