Linear Functions
Evaluating linear functions, building models from data, and reading graphs
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f(x) = mx + b. The SAT tests whether you can evaluate, interpret, build, and graph linear functions.
Why this matters
There are five question types here: plug in and compute, interpret slope and intercept in context, build the function from data, read a graph, and manipulate abstract definitions with unknown constants.
The five patterns
Direct Calculation
Plug an input into a linear function to find the output, or work backward from a given output to find the input. Order of operations is the main source of errors.
›Context & Models
Build a linear function from a word problem or explain what the slope and y-intercept mean in context. The coefficient touching the variable is the rate; the standalone number is the starting value.
›Data to Equation
Given a table, two points, or a slope with a point, write the equation f(x) = mx + b. Compute slope first, then back-solve for b using any known point.
›Graphical Analysis
Read intercepts, specific values, or slope interpretation from a graph of a linear function. Harder versions show a shifted graph like y = f(x) + k and ask you to recover f(x).
›Abstract Functions
Find an unknown constant, combine two functions, or evaluate a function at an expression like (k − 5). These test algebraic manipulation more than real-world reasoning.