Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Represent linear functions using tables, equations, and graphs.
• Understand and find the gradient (m) and y-intercept (c).
• Draw linear graphs using different methods.
• Identify and create tables for simple non-linear functions.
• Plot and draw simple non-linear graphs.

Lesson summary

Welcome to Week 9 of Grade 9 Mathematics! This week, we delve into the fascinating world of functions and graphs, focusing on linear functions and introducing simple non-linear functions. Understanding functions and graphs is a crucial skill in mathematics, forming the foundation for more advanced topics you'll encounter in higher grades. It's not just about numbers on a page; it's about understanding relationships and patterns that exist all around us. Think about the cost of airtime depending on how much you buy, or the distance a taxi travels over time. These are real-life scenarios that can be represented and understood using functions and graphs.

Lesson notes

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

We can think of it like a machine: you put in an x-value, and the machine processes it to give you a y-value. 2.

2. Linear Functions A linear function is a function whose graph is a straight line. The general equation 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 (slope) of the line – it tells us how steep the line is and whether it's increasing or decreasing. c is the y-intercept – the point where the line crosses the y-axis (the value of y when x = 0). 2.2.1 Understanding the Gradient (m) The gradient (m) is the "rise over run," meaning the change in y divided by the change in x. A positive gradient indicates an increasing line (going upwards from left to right), while a negative gradient indicates a decreasing line (going downwards from left to right). A gradient of zero means the line is horizontal.

Mathematically: m = (y₂ - y₁) / (x₂ - x₁) Where (x₁, y₁) and (x₂, y₂) are any two points on the line. 2.2.

2. Understanding the Y-intercept (c) The y-intercept (c) is the point where the line intersects the y-axis. Its coordinates are (0, c). This is the value of y when x is zero. 2.

3. Methods for Drawing Linear Graphs There are two common methods: Gradient-Intercept Method: Identify the gradient (m) and y-intercept (c) from the equation. Plot the y-intercept (0, c). Use the gradient to find another point. For example, if m = 2 (which can be written as 2/1), from the y-intercept, move 1 unit to the right and 2 units up, and plot the point. Draw a straight line through these two points.

Dual Intercept Method: Find the x-intercept (where the line crosses the x-axis, y = 0) and the y-intercept (where the line crosses the y-axis, x = 0). Plot these two points and draw a straight line through them. 2.

4. Simple Non-Linear Functions Unlike linear functions, non-linear functions do not produce straight-line graphs.

We'll focus on two simple types: Quadratic Functions (y = ax²): These functions produce a U-shaped curve called a parabola. The value of a determines whether the parabola opens upwards (if a > 0) or downwards (if a 0)**: These functions exhibit rapid growth or decay. If a > 1, the function grows exponentially (the graph increases sharply as x increases). If 0 3x = 6 => x =

2. Plot the point (2, 0).

Step 2: Find the y-intercept (x = 0): Substitute x = 0 into the equation: 3(0) + 2y = 6 => 2y = 6 => y =

3. Plot the point (0, 3).

Step 3: Draw the line: Draw a straight line through the points (2, 0) and (0, 3).

Example 3: Non-Linear Function - Quadratic Function Draw the graph of y = x² Step 1: Create a table of values: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | --- | -- | -- | -- | - | - | - | - | | y | 9 | 4 | 1 | 0 | 1 | 4 | 9 | Step 2: Plot the points: Plot the points from the table on a Cartesian plane.

Step 3: Draw a smooth curve: Draw a smooth U-shaped curve through the points.

Example 4: Non-Linear Function - Exponential Function Draw the graph of y = 2ˣ Step 1: Create a table of values: | x | -2 | -1 | 0 | 1 | 2 | 3 | | --- | ----- | ----- | - | - | - | - | | y | 0.25 | 0.5 | 1 | 2 | 4 | 8 | Step 2: Plot the points: Plot the points from the table on a Cartesian plane.

Step 3: Draw a smooth curve: Draw a smooth curve through the points. Note that the graph approaches the x-axis but never touches it. Guided Practice (With Solutions)

Question 1: Determine the gradient and y-intercept of the line y = -3x +

5. Solution: Comparing the equation to y = mx + c, we can see that m = -3 and c =

5. Gradient (m): -3 This means the line slopes downwards from left to right. For every 1 unit increase in x, y decreases by 3 units.

Y-intercept (c): 5 This means the line crosses the y-axis at the point (0, 5).

Question 2: Find the equation of a line with a gradient of 1/2 and a y-intercept of -

2. Solution: Using the equation y = mx + c, substitute m = 1/2 and c = -2: y = (1/2)x - 2 Question 3: Complete the table of values for the function y = x² - 1 and sketch the graph. | x | -2 | -1 | 0 | 1 | 2 | | --- | -- | -- | - | - | - | | y | | | | | | Solution: Calculate the y-values for each x-value: x = -2: y = (-2)² - 1 = 4 - 1 = 3 x = -1: y = (-1)² - 1 = 1 - 1 = 0 x = 0: y = (0)² - 1 = 0 - 1 = -1 x = 1: y = (1)² - 1 = 1 - 1 = 0 x = 2: y = (2)² - 1 = 4 - 1 = 3 Completed Table: | x | -2 | -1 | 0 | 1 | 2 | | --- | -- | -- | -- | - | - | | y | 3 | 0 | -1 | 0 | 3 | Sketch the graph: Plot the points (-2, 3), (-1, 0), (0, -1), (1, 0), and (2, 3) and draw a smooth U-shaped curve through them. This is a parabola. Independent Practice (Questions Only) Determine the gradient and y-intercept of the line y = 4x -

7. Find the equation of a line with a gradient of -2 and a y-intercept of

3. Draw the graph of the linear function y = -x + 2 using the gradient-intercept method.