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

y = mx + b

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)

m = (y₂ - y₁) / (x₂ - x₁)

Step 2: Use point-slope form

y - y₁ = m(x - x₁)

Step 3: Solve for y to get slope-intercept form

y = mx + b

Example: Find the Equation of a Line

Find the equation of the line passing through the points (2, 3) and (4, 7).

Step 1: Find the slope

m = (7 - 3) / (4 - 2) = 4 / 2 = 2

Step 2: Use point-slope form with (2, 3)

y - 3 = 2(x - 2)

y - 3 = 2x - 4

y = 2x - 4 + 3

y = 2x - 1

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.

C(x) = 5x + 1000

Temperature Conversion

The formula to convert Celsius to Fahrenheit is a linear function:

F = (9/5)C + 32

Practice Linear Functions

Test your knowledge with our interactive linear functions practice exercises.

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Linear Function Calculator

Use our calculator to graph linear functions and find equations.

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Key Concepts

1
A linear function has a constant rate of change (slope).
2
The graph of a linear function is always a straight line.
3
The slope-intercept form (y = mx + b) is the most common way to express a linear function.
4
Parallel lines have the same slope but different y-intercepts.
5
Perpendicular lines have slopes that are negative reciprocals of each other.

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