Lesson 3 of 5 · Algebra
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Linear functions are one of the most heavily tested Algebra topics and the language of most real-world SAT models. You’ll evaluate and interpret function notation, build from a table, graph or description, and explain what the slope and intercept mean in context (dollars per ticket, gallons per mile, degrees per thousand feet). Easy versions take 30 seconds. Hard versions hide the slope behind notation like , scaled units or a shifted input.
| Form | Typical stem | What it really tests |
|---|---|---|
| Evaluate or solve | “What is the value of ?” / “For what value of is ?” | Input vs. output |
| Build the function | “Which equation defines ?” (from a table, graph or description) |
3 questions on linear functions. Every option is explained once you answer.
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The first 3 of 10 cards for this topic. Sign in and finish the lesson to review them with spaced repetition.
What does mean on a graph?
| Slope and starting value |
| Interpret | “What is the best interpretation of 0.035 in this context?” / “…of the statement ?” | Units and meaning |
| Rate reasoning | “If increases by 9, by how much does increase?” |
| Parallel and perpendicular graphs | “The graph of is perpendicular to…” | Slope relationships |
About 30% of SAT Math questions are set in context, and linear models are among the most common. Expect both MCQ and SPR.
means the input produces the output . Equivalently, the point is on the graph of .
A linear function changes by equal differences over equal intervals. That’s how you recognize one in a table.
| Given | How to get | How to get |
|---|---|---|
| Two points or two function values | Substitute one point into | |
| A table | Change in output ÷ change in input (the -values may not go up by 1) | Step back to , or substitute |
| A graph | Rise over run between two lattice points | Read where the line crosses the -axis |
| A description | The rate (“per”, “each”, “every”), negative for decreases | The starting amount at the time the question calls 0 |
For linear , the intercept cancels whenever you look at a change:
A model such as is written in point-slope style. Its constant, 250, is the value when , not when . The same is true when the question redefines when time starts (“ years after 2014”): recompute the starting value for the new zero.
Graphs of linear functions are lines, so the usual rules apply. Parallel graphs have equal slopes. Perpendicular graphs have slopes whose product is , for example and .
Translate notation into points before anything else. “ and ” means the points and , and from there it’s a slope problem.
Interpretation questions: assign units, then use a template.
Any choice with the wrong units (miles per gallon instead of gallons per mile, or per hour when the input is miles) is out.
Test a middle or last row of a table. Distractors are often built to fit only the first row.
Desmos:
f(x)=..., then type f(6) or f(f(3)) to evaluate instantly.y1 ~ m x1 + b to get the exact and .Watch the zero point. Words like “after 2014”, “since the tank was filled” and “for ” define what the input measures and where it starts.
Pace. Evaluate or solve questions should take under 40 seconds. Hard interpretation questions deserve about 90 seconds, so read every choice against the units.
y1 ~ m x1 + b to fit a table exactly.y=f(x) and y=39 and click the intersection.