Linear Functions
What are Linear Functions?
A linear function is a function whose graph is a straight line. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
Standard Form of a Linear Function
m = slope
The slope tells us how steep the line is and whether it goes up or down as x increases.
b = y-intercept
The y-intercept is the point where the line crosses the y-axis (when x = 0).
Graphing Linear Functions
To graph a linear function, you can use the slope and y-intercept or plot points by creating a table of values.
Example: Graph y = 2x + 1
Method 1: Using Slope and Y-intercept
- Identify the y-intercept: b = 1, so plot the point (0, 1).
- Identify the slope: m = 2, which means "rise 2, run 1".
- From the y-intercept, move right 1 unit and up 2 units to find another point: (1, 3).
- Draw a line through these points.
Method 2: Table of Values
x | y = 2x + 1 | (x, y) |
---|---|---|
-1 | 2(-1) + 1 = -1 | (-1, -1) |
0 | 2(0) + 1 = 1 | (0, 1) |
1 | 2(1) + 1 = 3 | (1, 3) |
Finding the Equation of a Line
Given Two Points
To find the equation of a line passing through two points (x₁, y₁) and (x₂, y₂):
Step 1: Find the slope (m)
Step 2: Use point-slope form
Step 3: Solve for y to get slope-intercept form
Example: Find the Equation of a Line
Find the equation of the line passing through the points (2, 3) and (4, 7).
Applications of Linear Functions
Linear functions are used to model many real-world situations where one quantity changes at a constant rate relative to another.
Cost Functions
A company's cost function might be C(x) = 5x + 1000, where x is the number of items produced, 5 is the cost per item, and 1000 is the fixed cost.
Temperature Conversion
The formula to convert Celsius to Fahrenheit is a linear function:
Related Topics
Practice Linear Functions
Test your knowledge with our interactive linear functions practice exercises.
Start PracticeLinear Function Calculator
Use our calculator to graph linear functions and find equations.
Open CalculatorKey Concepts
- 1A linear function has a constant rate of change (slope).
- 2The graph of a linear function is always a straight line.
- 3The slope-intercept form (y = mx + b) is the most common way to express a linear function.
- 4Parallel lines have the same slope but different y-intercepts.
- 5Perpendicular lines have slopes that are negative reciprocals of each other.