How do you write a rule for a linear function?

You can write a rule for a linear function by determining its slope and y-intercept and plugging those values into the slope-intercept form. A linear function can be expressed as an equation.

Understanding Linear Functions

A linear function represents a straight line relationship between two variables. The most common way to write a linear function is using the slope-intercept form.

The Slope-Intercept Form

The slope-intercept form of a linear equation is:

y = mx + b

Where:

• y is the dependent variable (the output).
• x is the independent variable (the input).
• m is the slope of the line (the rate of change of y with respect to x).
• b is the y-intercept (the value of y when x = 0).

Steps to Write a Rule for a Linear Function

Here are the steps to write a rule for a linear function:

1. Determine the Slope (m): The slope represents how much y changes for every unit change in x. You can calculate the slope if you have two points (x₁, y₁) and (x₂, y₂) on the line using the following formula:

   m = (y₂ - y₁) / (x₂ - x₁)

2. Determine the Y-intercept (b): The y-intercept is the point where the line crosses the y-axis (where x = 0).

   • If you know the y-intercept directly: You have your b value.
   • If you don't know the y-intercept: Substitute the slope m and the coordinates of any point (x, y) on the line into the equation y = mx + b and solve for b.

3. Write the Equation: Substitute the values you found for m and b into the slope-intercept form y = mx + b.

Example

Let's say you have a line that passes through the points (1, 5) and (2, 8).

1. Find the slope (m):

   m = (8 - 5) / (2 - 1) = 3 / 1 = 3

2. Find the y-intercept (b):

   Using the point (1, 5) and the slope m = 3, substitute into the equation:

   5 = 3(1) + b
   5 = 3 + b
   b = 2

3. Write the equation:

   y = 3x + 2

Therefore, the rule for the linear function that passes through the points (1, 5) and (2, 8) is y = 3x + 2.

Summary

Element	Description
y	Dependent variable.
x	Independent variable.
m	Slope, calculated as (change in y) / (change in x).
b	Y-intercept, the point where the line crosses the y-axis (x=0).
Equation Form	y = mx + b