Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Represent linear functions as y = mx + c
• Sketch graphs of linear functions
• Identify features like intercepts and gradient
• Recognize simple non-linear functions (e.g., y = x², y = 1/x)
• Create tables of values to plot non-linear graphs

Lesson summary

This week, we delve into the fascinating world of functions and graphs, focusing specifically on linear functions and introducing some simple non-linear functions. Understanding functions and graphs is crucial because they are powerful tools for representing and analyzing relationships between quantities. From tracking cellphone data usage to predicting population growth, functions are everywhere! In South Africa, understanding graphs can help us interpret socio-economic trends, analyse the spread of diseases, or even understand the performance of our favourite sports teams.

Lesson notes

2.1 Linear Functions: The Straight Story 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 (usually plotted on the vertical axis). x is the independent variable (usually plotted on the horizontal axis). m is the slope or gradient of the line. It represents the rate of change of y with respect to x. A positive slope indicates an increasing line (going upwards from left to right), a negative slope indicates a decreasing line, a slope of 0 indicates a horizontal line, and an undefined slope (division by zero) indicates a vertical line. The slope can be calculated using two points on the line (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁) c is the y-intercept. It is the point where the line crosses the y-axis (i.e., the value of y when x = 0).

Example 1: Understanding Slope and Y-Intercept Consider the equation: y = 2x + 3 Slope (m):

2. This means for every increase of 1 in x, y increases by

2. Y-intercept (c):

3. The line crosses the y-axis at the point (0, 3).

Example 2: Finding the Equation of a Line Find the equation of a line that passes through the points (1, 5) and (3, 9).

Step 1: Calculate the slope (m). m = (9 - 5) / (3 - 1) = 4 / 2 = 2 Step 2: Use the slope and one of the points to find the y-intercept (c). Using the point (1, 5) and the equation y = mx + c: 5 = 2(1) + c 5 = 2 + c c = 3 Step 3: Write the equation of the line. y = 2x + 3 Example 3: Graphing a Linear Function Graph the linear function y = -x + 4 Method 1: Using Slope and Y-intercept Y-intercept (c):

4. Plot the point (0, 4) on the y-axis.

Slope (m): -

1. From the y-intercept, move 1 unit to the right and 1 unit down. Plot the new point (1, 3). Draw a straight line through the two points.

Method 2: Using Two Points Choose two values for x, and calculate the corresponding y values. If x = 0, y = -0 + 4 =

4. Point (0, 4) If x = 4, y = -4 + 4 =

0. Point (4, 0) Plot the two points (0, 4) and (4, 0) and draw a straight line through them. 2.2 Simple Non-Linear Functions: A Taste of Curves Non-linear functions are functions whose graphs are not straight lines.

We will focus on two simple examples: Quadratic Function: y = x² This function creates a parabola shape. The graph is symmetrical around the y-axis. As x increases (either positively or negatively), y increases rapidly.

Reciprocal Function: y = 1/x This function creates a hyperbola shape. The graph has two separate branches. As x approaches 0, y approaches infinity (or negative infinity). The graph has asymptotes at x=0 and y=

0. Example 4: Plotting y = x² Create a table of values and plot the graph of y = x² for x values from -3 to 3. | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | ---- | --- | --- | -- | - | - | - | - | | y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 | Plot these points on a graph and connect them with a smooth curve. The resulting shape is a parabola.

Example 5: Plotting y = 1/x Create a table of values and plot the graph of y = 1/x for x values from -3 to -0.5 and from 0.5 to

3. Note that x cannot be 0. | 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 | Plot these points on a graph and connect them with smooth curves. The resulting shape is a hyperbola with two branches. Guided Practice (With Solutions)

Question 1: Determine the equation of a straight line that passes through the points (0, -2) and (2, 4).

Solution: Step 1: Find the slope (m). m = (4 - (-2)) / (2 - 0) = 6 / 2 = 3 Step 2: Find the y-intercept (c). Since the point (0, -2) is given, we know that the y-intercept is -

2. So, c = -

2. Step 3: Write the equation of the line. y = 3x - 2

Commentary: This question tests the ability to calculate the slope and identify the y-intercept from two points. Recognizing that (0, -2) directly provides the y-intercept simplifies the process.

Question 2: Sketch the graph of the linear function y = -0.5x +

1. Label the intercepts.

Solution: Step 1: Find the y-intercept. The y-intercept is

1. Plot the point (0, 1).

Step 2: Find the x-intercept. To find the x-intercept, set y = 0 and solve for x: 0 = -0.5x + 1 5x = 1 x = 2 Plot the point (2, 0).

Step 3: Draw a straight line through the points (0, 1) and (2, 0). Label the y-intercept as (0, 1) and the x-intercept as (2, 0) on your graph.

Commentary: This question requires understanding how to find both intercepts and then use them to accurately sketch the line. Pay close attention to the negative slope, which indicates a decreasing line.

Question 3: For the function y = x², find the value of y when x = -4 and when x =

4. What does this tell you about the symmetry of the graph?

Solution: When x = -4: y = (-4)² = 16 When x = 4: y = (4)² = 16 This tells us that the graph is symmetrical about the y-axis because the same y-value (16) is obtained for both x = -4 and x = 4.