Goals
1. Understand the definition and characteristics of linear functions.
2. Identify and describe the domain and range of linear functions.
3. Apply the concepts of linear functions to practical problems.
Contextualization
Linear functions are essential mathematical tools that help us model and analyse various everyday situations and scenarios in the job market. For example, they are used to calculate costs and forecast profits in businesses, estimate population growth in urban planning, and analyse investment returns in finance. Grasping how these functions work and being able to interpret their graphs is crucial for making informed and effective decisions across various fields.
Subject Relevance
To Remember!
Definition of Linear Function
A linear function is a mathematical function that can be represented by the formula f(x) = ax + b, where 'a' and 'b' are constants and 'x' is the independent variable. The key feature of this function is that its graph forms a straight line.
-
The constant 'a' is known as the slope coefficient and it determines how steep the line is.
-
The constant 'b' is referred to as the y-intercept and indicates where the line crosses the y-axis.
-
These functions are commonly used to model direct linear relationships between two variables.
Identification of Domain and Range
The domain of a linear function consists of all possible values for the independent variable x, while the range encompasses all potential output values of the function.
-
For linear functions, the domain typically includes all real numbers.
-
The range also consists of all real numbers since a straight line can take any value along the y-axis.
-
Identifying the domain and range is essential for comprehending the behaviour of the function.
Graphical Representation of Linear Functions
The graphical representation of a linear function is a straight line on a Cartesian plane. The slope of the line and the point where it intersects the y-axis are determined by the slope and y-intercept coefficients, respectively.
-
The graph of f(x) = ax + b is a straight line with slope 'a' and y-intercept 'b'.
-
To draw the graph, find two points that satisfy the equation and connect them with a line.
-
The slope indicates whether the function is increasing (a > 0) or decreasing (a < 0).
Practical Applications
-
Profit forecasting: Businesses use linear functions to predict their profits based on past sales data.
-
Cost analysis: Linear functions assist in calculating the total production cost of a product, taking both fixed and variable costs into account.
-
Population growth: Urban planners apply linear functions to model and forecast the population growth of a city.
Key Terms
-
Linear Function: A function represented by the formula f(x) = ax + b.
-
Slope Coefficient: The constant 'a' in a linear function, which defines the line's slope.
-
Y-Intercept: The constant 'b' in a linear function, which defines where the line intersects the y-axis.
-
Domain: The set of all possible values for the independent variable x.
-
Range: The set of all potential output values of the function.
Questions for Reflections
-
How can linear functions be used to address everyday challenges?
-
In what way can mastering linear functions enhance your future career prospects?
-
What are the limitations of linear functions when it comes to modelling real-world situations?
Modeling Population Growth
In this mini-challenge, you will use linear functions to model the population growth of a city.
Instructions
-
Form groups of 4 to 5 students.
-
Choose a fictional or real city to model its population growth.
-
Research historical population growth data for the chosen city, if available.
-
Use this data to create a linear function representing population growth.
-
Identify and describe the domain and range of your function.
-
Graph the function and discuss the implications of your model with your group.
-
Prepare a brief presentation to share your findings with the class.
