Linear Functions — SAT Math Explained
A linear function is a relationship between two variables where the change in the output is constant for every unit change in the input, always producing a straight-line graph. Written as f(x) = mx + b or y = mx + b.
The Core Idea
Functions describe how one quantity depends on another. In a linear function, that dependency is perfectly proportional — equal increases in input always produce equal increases in output.
Key Vocabulary
A rule where each input has exactly one output
The independent variable — you choose it
The dependent variable — it's determined by the function
Describes a relationship that graphs as a straight, non-curved line
Read 'f of x' — another way to write y; emphasizes that the output depends on x
Properties
The slope never changes — equal steps in x always produce equal steps in y
A linear function always graphs as a straight line, never curved
For every x value, there is exactly one y value
All real numbers — you can plug in any x
All real numbers (unless domain is restricted)
How To Identify Linear Functions
Equation has no exponents on variables other than 1
No variables multiplied together
Graph is a straight line
Table of values shows a constant rate of change
Real World Examples
A taxi that charges $3 per mile: cost = 3x
Temperature conversion: F = 1.8C + 32
Savings account growing by $50/month: balance = 50t + initial
Phone plan: monthly bill = 0.10 × minutes + $20 base
How This Connects to Other Topics
Directly leads into slope, slope-intercept form, graphing, and systems of equations
Practice: Linear Functions
5 SAT-style questions. Select your answer and get an instant explanation.
Which of the following represents a linear function?
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