Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Define linear, quadratic, and exponential functions.
• Sketch graphs of linear functions.
• Identify intercepts and gradients of linear functions.
• Generate a table of values for non-linear functions to sketch them.
• Interpret basic features of quadratic and exponential graphs.

Lesson summary

Functions and graphs are fundamental tools in mathematics, providing a visual and algebraic way to understand relationships between variables. This week, we'll build on your previous knowledge of linear functions and expand into exploring simple non-linear relationships. Understanding these concepts is crucial because they form the basis for more advanced mathematics and are widely applied in various fields like economics, science, and engineering. For example, understanding how the price of electricity (a function of usage) relates to your family's monthly bill is a real-life application. Similarly, understanding population growth or decay often involves non-linear functions.

Lesson notes

2.1 What is a Function? A function is a special type of relationship between two variables, usually denoted as x and y. For every value of x (the input or independent variable), there is only one corresponding value of y (the output or dependent variable). We often write this relationship as y = f(x), where f is the function's name. Think of a function like a machine. You put something in (x), and the machine does something to it and gives you something out (y). For example, if f(x) = 2x + 1, then if you put in x = 3, the machine calculates 2(3) + 1 = 7, so y = 7. 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: m is the gradient (or slope) of the line. It tells you how steep the line is and whether it goes up or down as you move from left to right. A positive m means the line goes up (increasing function), and a negative m means the line goes down (decreasing function). c is the y-intercept. This is the point where the line crosses the y-axis (where x = 0).

Example 1: Consider the linear function y = 3x -

2. The gradient, m, is

3. This means for every 1 unit you move to the right on the graph, you move 3 units up. The y-intercept, c, is -

2. This means the line crosses the y-axis at the point (0, -2). 2.3 Sketching Linear Functions There are several ways to sketch a linear function: Using a table of values: Choose a few values for x, calculate the corresponding values for y, plot the points on a graph, and draw a straight line through them.

Using the slope-intercept form: Plot the y-intercept (0, c). Then, use the gradient m to find another point on the line. Remember, m is rise/run. For example, if m = 2 (or 2/1), start at the y-intercept, move 1 unit to the right (run), and then 2 units up (rise). Plot this new point and draw a line through both points.

Example 2: Sketching y = -2x + 1 using a table of values: | x | y = -2x + 1 | | --- | ----------- | | -1 | 3 | | 0 | 1 | | 1 | -1 | | 2 | -3 | Plot the points (-1, 3), (0, 1), (1, -1), and (2, -3) on a graph and draw a straight line through them. 2.4 Simple Non-Linear Functions: Quadratic Functions A quadratic function is a function of the form: y = ax² + bx + c Where a, b, and c are constants, and a ≠

0. The graph of a quadratic function is a parabola (a U-shaped curve). If a > 0, the parabola opens upwards (it has a minimum turning point). If a (x - 2)(x + 2) = 0 => x = 2 or x = -

2. The x-intercepts are (2, 0) and (-2, 0). To find the y-intercept, set x = 0: y = 0² - 4 = -

4. The y-intercept is (0, -4). By observing the graph, or using the formula x = -b/2a (where b=0 here), we can see the axis of symmetry is x = 0 and the turning point is (0, -4). 2.5 Simple Non-Linear Functions: Exponential Functions An exponential function is a function of the form: y = ab x Where a and b are constants, a ≠ 0, and b > 0, b ≠

1. If b > 1, the function represents exponential growth. As x increases, y increases rapidly. Think of compound interest or population growth. If 0 x . | x | y = 2 x | | --- | ----------- | | -2 | 0.25 | | -1 | 0.5 | | 0 | 1 | | 1 | 2 | | 2 | 4 | | 3 | 8 | Plotting these points reveals exponential growth. As x gets larger, y increases dramatically.

Example 5: Consider the exponential function y = (1/2) x . This represents exponential decay. 2.6 Using a Table of Values to Sketch Non-Linear Graphs Creating a table of values is a crucial step in sketching non-linear graphs. Choose a range of x values (positive, negative, and zero) and calculate the corresponding y values. Plot these points carefully and connect them with a smooth curve. The more points you plot, the more accurate your sketch will be. Guided Practice (With Solutions)

Question 1: Identify the gradient and y-intercept of the linear function y = -x +

5. Sketch the graph.

Solution: The gradient, m, is -1 (remember -x is the same as -1x). The y-intercept, c, is

5. This means the line crosses the y-axis at (0, 5). To sketch the graph, plot the point (0, 5). Since the gradient is -1, from (0,5) move 1 unit to the right and 1 unit down to find another point (1, 4). Draw a line through (0, 5) and (1, 4).

Question 2: Generate a table of values for y = x² + 1 for x values from -2 to

2. Sketch the graph. What kind of graph is this?

Solution: | x | y = x² + 1 | | --- | ----------- | | -2 | 5 | | -1 | 2 | | 0 | 1 | | 1 | 2 | | 2 | 5 | Plot the points (-2, 5), (-1, 2), (0, 1), (1, 2), and (2, 5). Connect them with a smooth U-shaped curve. This is a quadratic graph (parabola).

Question 3: The price of a new car depreciates at approximately 15% per year. If the car initially costs R200,000, write an equation that models the car's value (V) after t years. What type of function is this?

Solution: The equation is V = 200000(0.85) t . (Since it depreciates by 15% each year, the remaining value is 100% - 15% = 85% = 0.85). This is an exponential decay function. Note that 0 x *. What is the y-intercept?