Slope and Y-Intercept
The Complete SAT Math Guide — From Fundamentals to Mastery
Master every aspect of linear equations: slope, y-intercept, graphing, parallel & perpendicular lines, equation forms, and real-world applications.
Part 1: Fundamental Concepts
What Is a Linear Relationship?
A linear relationship is a relationship between two variables where the graph is a straight line. It has a constant rate of change — meaning for every equal increase in $x$, the value of $y$ changes by the same amount.
The Slope-Intercept Form
$$y = mx + b$$
- $m$ = slope (steepness and direction of the line)
- $b$ = y-intercept (where the line crosses the y-axis)
- $y$ is isolated on the left side
Why Does This Form Matter on the SAT?
Part 2: Slope ($m$) — The Complete Guide
Definition of Slope
Slope measures the steepness and direction of a line. It tells you how much $y$ changes when $x$ increases by 1.
The Slope Formula
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}}$$
Rise and Run
- Rise = vertical change (up is positive, down is negative)
- Run = horizontal change (right is positive, left is negative)
- Slope = rise ÷ run
Four Types of Slope
Positive Slope
↗
$m > 0$
Line goes up from left to right
As $x$ increases, $y$ increases
Negative Slope
↘
$m < 0$
Line goes down from left to right
As $x$ increases, $y$ decreases
Zero Slope
→
$m = 0$
Horizontal line
Equation: $y = b$
Undefined Slope
↑
run = 0
Vertical line
Equation: $x = a$
Interactive: See All Four Types of Slope
Slope Magnitude (Steepness)
The absolute value of the slope determines steepness:
- Larger $|m|$ → steeper line ($|m| = 3$ is steeper than $|m| = 1$)
- Smaller $|m|$ → flatter line (closer to horizontal)
- $|m| = 1$ → line makes a 45° angle with the x-axis
Slope as a Rate of Change
In real-world problems, slope represents a rate — "$y$ per $x$":