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.
• Identify the gradient and y-intercept.
• Sketch graphs of linear functions.
• Identify simple non-linear relationships (parabolas and hyperbolas).
• Plot simple non-linear graphs from a table.

Lesson summary

This week, we delve into the fascinating world of functions and graphs. Understanding functions and how they are represented graphically is a fundamental skill in mathematics. It allows us to model real-world relationships and make predictions based on these models. Imagine planning a stokvel where each member contributes a fixed amount monthly. The total amount in the stokvel is a function of the number of members. Or, think about calculating the cost of data; the more data you use, the higher your bill. Understanding functions helps you predict your costs.

Lesson notes

2.1 What is a Function? A function is a special relationship between two sets of numbers (or variables) where each input has ONLY one output.

Think of it like a machine: you put something in (the input), and it spits out something else (the output). We usually call the input 'x' and the output 'y'.

Input (x): Also known as the independent variable.

Output (y): Also known as the dependent variable because its value depends on the value of x. We often write y as f(x), which means "y is a function of x". 2.2 Linear Functions A linear function is a function whose graph is a straight line.

The general form of a linear equation is: `y = mx + c` `m` is the gradient (or slope) of the line. It tells us how steep the line is and whether it's increasing (positive `m`) or decreasing (negative `m`). Gradient = (Change in y) / (Change in x) = (y2 - y1) / (x2 - x1) `c` is the y-intercept. It's the point where the line crosses the y-axis (where x = 0).

Example 1: Linear Function Consider the equation: `y = 2x + 1` `m = 2`: For every 1 unit increase in x, y increases by 2 units. `c = 1`: The line crosses the y-axis at the point (0, 1). To plot this, we can choose two points: If `x = 0`, then `y = 2(0) + 1 = 1`.

Point: (0, 1) If `x = 1`, then `y = 2(1) + 1 = 3`.

Point: (1, 3) Plot these two points on a graph and draw a straight line through them.

Example 2: Finding the equation of a line given two points. Suppose you have the points (1, 5) and (3, 9).

Calculate the gradient (m): m = (9 - 5) / (3 - 1) = 4 / 2 = 2 Use the gradient and one of the points to find c: Using the point (1, 5) and the equation y = mx + c, we have: 5 = 2(1) + c 5 = 2 + c c = 3 Write the equation: Therefore, the equation of the line is y = 2x + 3. 2.3 Simple Non-Linear Functions Unlike linear functions, non-linear functions do not form a straight line when graphed.

We will focus on two common types: Quadratic Functions: These have the general form `y = ax² + bx + c`, where `a`, `b`, and `c` are constants and `a ≠ 0`. Their graphs are parabolas (U-shaped curves).

Inverse Proportion (Hyperbola): These functions have the form `y = k/x`, where `k` is a constant. Their graphs are hyperbolas, which have two separate branches.

Example 3: Quadratic Function Consider the equation: `y = x² - 4` To plot this, let's create a table of values: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :--- | :-- | :-- | :-- | :- | :- | :- | :- | | y | 5 | 0 | -3 | -4 | -3 | 0 | 5 | Plot these points on a graph and connect them with a smooth curve to form a parabola. Notice the U-shape and the symmetry around the vertical line x =

0. Example 4: Inverse Proportion (Hyperbola)

Consider the equation: `y = 6/x` To plot this, let's create a table of values: | x | -3 | -2 | -1 | 1 | 2 | 3 | | :--- | :-- | :-- | :-- | :- | :- | :- | | y | -2 | -3 | -6 | 6 | 3 | 2 | Plot these points on a graph and connect them with smooth curves. Note that as x approaches zero from the positive side, y becomes very large and positive. As x approaches zero from the negative side, y becomes very large and negative. The graph consists of two separate curves. 2.4 Identifying Functions from Tables Linear: Look for a constant difference in `y` for every constant difference in `x`.

Quadratic: The second differences are constant. Calculate the first differences (the difference between consecutive y-values). Then calculate the difference between consecutive first differences. If these second differences are constant, the function is quadratic.

Inverse Proportion: Check if the product of `x` and `y` is constant. If `x y` is the same for all points in the table, it represents inverse proportion. Guided Practice (With Solutions)

Question 1: A cellphone company charges R1 per minute for calls plus a R5 connection fee. Write an equation to represent the total cost (y) as a function of the number of minutes (x). Draw the graph of this equation.

Solution: The equation is `y = 1x + 5` (or `y = x + 5`). The gradient (m) is 1, representing the R1 cost per minute. The y-intercept (c) is 5, representing the R5 connection fee.

To draw the graph: When x = 0 (no minutes used), y =

5. Plot the point (0, 5). When x = 5 (5 minutes used), y = 5 + 5 =

1

0. Plot the point (5, 10). Draw a straight line through these two points.

Question 2: Given the following table of values, determine if the function is linear, quadratic, or inverse proportion: | x | 1 | 2 | 3 | 4 | | :--- | :-- | :-- | :-- | :-- | | y | 3 | 12 | 27 | 48 | Solution: First Differences: 12 - 3 = 9; 27 - 12 = 15; 48 - 27 =

2

1. These are NOT constant, so it's not linear.

Second Differences: 15 - 9 = 6; 21 - 15 =

6. These ARE constant, so the function is quadratic.

Question 3: Sketch the graph of `y = -2x + 4`. Identify the gradient and y-intercept.

Solution: Gradient (m): -2 (The line slopes downwards).

Y-intercept (c): 4 (The line crosses the y-axis at (0, 4)).

To sketch: Plot the y-intercept (0, 4).