Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Identify linear, quadratic, and exponential relationships.
• Represent real-life scenarios with tables, graphs, and equations.
• Determine the equation of a linear relationship.
• Use these tools to make predictions and solve problems.
• Explain the meaning of key features on a graph in context.

Lesson summary

This week, we delve deeper into understanding how patterns, relationships, and representations show up in our everyday lives. Mathematical Literacy isn't just about numbers; it's about making sense of the world around us using mathematical tools. Understanding patterns, relationships and representations is crucial for making informed decisions about budgeting, understanding data presented in the news, interpreting graphs, and predicting future trends. Imagine needing to understand the impact of inflation on your family's budget, or interpreting the results of a COVID-19 infection rate graph – that's Mathematical Literacy in action.

Lesson notes

This section will cover three fundamental types of relationships: Linear, Quadratic, and Exponential. For each, we will explore how they are represented in tables, graphs, and equations, along with relevant examples applicable to the South African context.

A. Linear Relationships Definition: A linear relationship exists when there is a constant rate of change between two variables. This means for every fixed increase in one variable (the independent variable, usually 'x'), the other variable (the dependent variable, usually 'y') increases or decreases by a constant amount.

Table Representation: In a table, a linear relationship shows a constant difference between consecutive y-values for equal intervals of x-values. | x | y | |---|---| | 0 | 5 | | 1 | 8 | | 2 | 11| | 3 | 14| Notice that as 'x' increases by 1, 'y' increases by

3. This constant difference indicates a linear relationship.

Graphical Representation: A linear relationship is represented by a straight line on a graph. The slope (or gradient) of the line indicates the rate of change. A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing relationship.

Equation Representation: The general equation of a linear relationship is `y = mx + c` or `y = a + bx`, where: `m` (or `b`) is the slope (gradient) – the rate of change. `c` (or `a`) is the y-intercept – the value of 'y' when 'x' is zero.

Example 1: Cell Phone Costs A Vodacom user pays a monthly subscription fee of R100 and R0.50 per megabyte (MB) of data used. Represent this relationship in a table, graph, and equation.

Table: | Data Used (MB) (x) | Total Cost (R) (y) | |---|---| | 0 | 100 | | 100 | 150 | | 200 | 200 | | 300 | 250 | Equation: The equation is `y = 0.50x + 100`, where 'x' is the data used in MB and 'y' is the total cost in Rands.

Graph: (Description: A straight line starting at y=100 on the y-axis, with a positive slope.) You would plot the points from the table on a graph, with data usage (x) on the horizontal axis and cost (y) on the vertical axis. Draw a line through the points.

B. Quadratic Relationships Definition: A quadratic relationship is a relationship where one variable is related to the square of another variable.

Table Representation: In a table, the second difference between consecutive y-values is constant for equal intervals of x-values. This is a tell-tale sign of a quadratic relationship. | x | y | First Difference | Second Difference| |---|---|---|---| | 0 | 0 | | | | 1 | 1 | 1 | | | 2 | 4 | 3 | 2 | | 3 | 9 | 5 | 2 | | 4 | 16| 7 | 2 | The first difference is found by subtracting each y value from the following y value (e.g. 1-0 = 1, 4-1 = 3). The second difference is found by subtracting each first difference value from the following first difference value (e.g. 3-1 = 2, 5-3 = 2).

Graphical Representation: A quadratic relationship is represented by a parabola – a U-shaped curve. The parabola can open upwards (if the coefficient of x 2 is positive) or downwards (if the coefficient of x 2 is negative). It has a maximum or minimum point called the vertex (turning point).

Equation Representation: The general equation of a quadratic relationship is `y = ax² + bx + c`, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

Example 2: Projectile Motion (Simplified) A ball is thrown upwards. Its height (in meters) above the ground after 't' seconds is approximated by the equation: `h = -5t² + 20t`.

Table: | Time (t) seconds | Height (h) meters | |---|---| | 0 | 0 | | 1 | 15 | | 2 | 20 | | 3 | 15 | | 4 | 0 | Graph: (Description: A parabola opening downwards, with the vertex at (2,20))

Interpretation: The negative coefficient of the `t²` term indicates that the parabola opens downwards. The maximum height of the ball is 20 meters, reached after 2 seconds.

C. Exponential Relationships Definition: An exponential relationship exists when the rate of change is proportional to the current value. This often represents growth or decay that accelerates over time.

Table Representation: In a table, the y-values are multiplied by a constant factor for equal intervals of x-values. | x | y | |---|---| | 0 | 2 | | 1 | 6 | | 2 | 18| | 3 | 54| Notice that as 'x' increases by 1, 'y' is multiplied by

3. This constant multiplier indicates an exponential relationship.

Graphical Representation: An exponential relationship is represented by a curve that gets increasingly steep (growth) or increasingly shallow (decay).

Equation Representation: The general equation of an exponential relationship is `y = a b^x`, where: `a` is the initial value (the value of 'y' when 'x' is 0). `b` is the growth/decay factor. If b > 1, it's growth. If 0 < b < 1, it's decay. `x` is the independent variable.

Example 3: Savings with Compound Interest Thando invests R1000 in a savings account that earns 8% compound interest per year. Represent this relationship in a table, graph, and equation.