Slope and Rate of Change — SAT Math Explained
Slope measures the steepness and direction of a line — it is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Mathematically: m = (y₂ - y₁) / (x₂ - x₁).
The Core Idea
Slope tells you how fast something is changing. A large slope means steep — big change in output for small change in input. A small slope means gradual. The sign tells you the direction.
Key Vocabulary
The vertical change between two points (change in y)
The horizontal change between two points (change in x)
Line goes upward left to right — as x increases, y increases
Line goes downward left to right — as x increases, y decreases
Perfectly horizontal line — y never changes
Perfectly vertical line — x never changes (division by zero)
Slope Interpretations
For every 1 unit increase in x, y increases by 2
For every 1 unit increase in x, y decreases by 0.5
y stays constant no matter what x does
If a car travels 60 miles per hour, slope = 60 on a distance-vs-time graph
How To Calculate
From two points (x₁, y₁) and (x₂, y₂): use m = (y₂ - y₁)/(x₂ - x₁)
From a graph: count rise over run between two clear lattice points
From slope-intercept form y = mx + b: m is the coefficient of x
From a table: divide the change in y by the change in x between any two rows
Real World Application
Slope = speed on a distance-time graph, slope = rate of production in manufacturing, slope = pitch of a roof
Common Errors to Avoid
Switching rise and run (dividing x-change by y-change instead of the reverse)
Forgetting the negative sign when the line falls from left to right
Using non-consistent point order (mixing up which is point 1 vs. point 2)
Practice: Slope and Rate of Change
5 SAT-style questions. Select your answer and get an instant explanation.
What is the slope of a line passing through (2, 4) and (6, 12)?
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