Linear Equations: Slope-Intercept Form

Rewrite linear equations in the form $y = ax + b$ , determine the slope $a$ , and find the y-intercept $b$ .

Solved: 0

Slope-Intercept Form & Function Notation

A linear equation can be written in multiple forms. One of the most commonly used forms is the slope-intercept form:

Linear Equation (Slope-Intercept Form)

y = ax + b

The slope-intercept form of a linear equation, where a represents the slope (also called the gradient) and b represents the y-intercept.

where:
- $a$ is the slope (also called the gradient), which describes the direction and steepness of the line.
- $b$ is the y-intercept, representing the y-coordinate of the point where the line crosses the y-axis, located at

(0, b)

.

Note: We will cover the slope in detail in the next topic.

Function Notation:
Once we isolate

y

in a linear equation (writing it as

y = ax + b

), we can express

y

in function notation as

y = f(x)

.

In our case, the linear equation

y = ax + b

becomes:

f(x) = ax + b

Using function notation is very useful. For example, it allows us to evaluate the function for specific values of

x

. To find the y-value when

x = 0

, we substitute

0

for

x

f(0) = a \cdot 0 + b = b

This shows that when

x = 0

, the function value is

b

, which corresponds directly to the y-intercept at the point

(0, b)

Learning Topics

Frequently Asked Questions

What is the slope-intercept form of a linear equation?

The slope-intercept form is $y = ax + b$, where **$a$ is the slope** and **$b$ is the y-intercept**.

How do we express $y$ as a function of $x$?

When we rewrite a linear equation to isolate $y$ in the form $y = ax + b$, the value of $y$ is determined entirely by $x$. This allows us to use function notation and write $y = f(x)$.