Linear Functions
Definition
A linear function is a mathematical expression that represents a relationship between two variables,
say 𝑥 and 𝑦, where the rate of change (slope) between them is constant. Linear functions are graphically represented as straight lines in a Cartesian plane.
The general form of a linear function is represented as:
\[ f(x) = mx + b \] where:
- \(m\): the slope of the line.
- \(b\): the y-intercept of the line.
This form is commonly referred to as the slope-intercept form.
It is graphically represented as a straight line in a Cartesian plane.
Examples of Linear Functions
Increasing Function:
An increasing linear function has a positive slope (𝑚>0), meaning the line rises as 𝑥 increases.
Example: \[ f(x) = 2x + 3 \]
For this function:
- Slope ( 𝑚) = 2 , which is positive.
- Graphically, the line moves upward from left to right.
The line rises as \(x\) increases.
Decreasing Function:
A decreasing linear function has a negative slope (𝑚< 0), meaning the line falls as 𝑥 increases
Example: \[ f(x) = -3x + 4 \]
For this function:
- Slope (𝑚 ) = − 3 , which is negative.
- Graphically, the line moves downward from left to right.
The line falls as \(x\) increases.
Constant Function:
A constant linear function has a slope of zero (𝑚 = 0), meaning the line is horizontal and does not change with 𝑥.
Example: \[ f(x) = 5 \]
For this function:
- Slope ( 𝑚 ) = 0 , so the line is flat.
- The function is a horizontal line at 𝑦 = 5 .
The line remains flat and constant at \(y = 5\).