2-6-4 Linear Functions Explained
Key Concepts of Linear Functions
Linear functions are fundamental in algebra, representing a straight line on a coordinate plane. The key concepts include:
- Definition of Linear Functions: Functions of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Slope: The rate at which the function increases or decreases.
- Y-Intercept: The point where the function crosses the y-axis.
- Graphing Linear Functions: Plotting points and drawing a straight line.
- Applications of Linear Functions: Real-world scenarios where linear functions are used.
1. Definition of Linear Functions
A linear function is a polynomial function of the first degree, represented by the equation \( f(x) = mx + b \). Here, \( m \) is the slope, which determines the steepness of the line, and \( b \) is the y-intercept, which is the value of \( y \) when \( x = 0 \).
Example:
Consider the linear function \( f(x) = 2x + 3 \). Here, the slope \( m \) is 2, and the y-intercept \( b \) is 3.
2. Slope
The slope of a linear function indicates how much the function changes for each unit change in \( x \). A positive slope means the function increases as \( x \) increases, while a negative slope means the function decreases.
Example:
For the function \( f(x) = 3x - 1 \), the slope \( m \) is 3, indicating that the function increases by 3 units for every 1 unit increase in \( x \).
3. Y-Intercept
The y-intercept is the point where the graph of the function crosses the y-axis. It is found by setting \( x = 0 \) in the function equation.
Example:
For the function \( f(x) = -4x + 5 \), the y-intercept \( b \) is 5, meaning the graph crosses the y-axis at the point (0, 5).
4. Graphing Linear Functions
To graph a linear function, plot the y-intercept and use the slope to find additional points. Connect these points to form a straight line.
Example:
Graph the function \( f(x) = \frac{1}{2}x - 2 \):
1. Plot the y-intercept (0, -2).
2. Use the slope \( \frac{1}{2} \) to find additional points: for each 1 unit increase in \( x \), \( y \) increases by 0.5.
3. Connect the points to draw the line.
5. Applications of Linear Functions
Linear functions are used in various real-world applications, such as calculating distance, predicting trends, and analyzing economic data.
Example:
A car rental company charges $50 per day plus $0.20 per mile. The total cost \( C \) in dollars for renting the car for \( d \) days and driving \( m \) miles can be modeled by the linear function \( C = 50d + 0.20m \).
Examples and Analogies
To better understand linear functions, consider the following analogy:
Imagine a linear function as a staircase. The slope represents the steepness of the stairs, and the y-intercept is the starting point at the bottom. Each step you take (increase in \( x \)) corresponds to a specific height gain (increase in \( y \)).
Practical Applications
Understanding linear functions is crucial for solving real-world problems, such as calculating costs, predicting future values, and analyzing trends. These skills are essential in fields like economics, engineering, and physics.