SAT Math — linear functions: rate of change and starting value

Almost every question in the Algebra domain is, underneath, the same question: which number here is the rate, and which is the value you begin from. A learner who can answer that reads a word problem, a table and a pair of coordinates as three views of one thing. A learner who cannot will solve the arithmetic correctly and still put the rate where the starting value belongs, which is the single commonest way marks are lost on this part of the test.

This set works on that one idea from six directions, and deliberately leaves systems, inequalities and quadratics to the other materials in the category.

Three questions give a table of values written out in a sentence — when the input is two the output is eleven, when it is five the output is twenty-three — and ask for the rate, for the value at zero, or for a value the table does not list. Working back to zero is where the arithmetic bites: the amount you take away is not the answer, and a learner who stops one step early gets a number that feels finished.

Three more give two points on a line and ask for the slope or for the whole equation. The wrong options here are built from the mistakes that actually happen: the fraction turned upside down, the two differences subtracted in opposite directions so the sign flips, a coordinate used in place of a difference. One of them is a line that passes through the second point perfectly well and misses the origin, which is a useful reminder that an equation has to satisfy both points rather than whichever one you tested.

Two questions use function notation, evaluating a function at a number and solving for the input that produces a given output, with distractors that stop at the intermediate step or reverse the order of the two operations.

Two ask only what a number means, with nothing to calculate: the flat fee in a taxi fare, the draining rate of a tank. Both are set up so that the tempting wrong reading is the other number in the same equation. And two ask for the model itself to be built from a situation in words, where the trap is attaching the recurring charge to the wrong quantity — checking the equation at zero catches it immediately, and the explanations say so.

Every explanation names the specific error behind each wrong option rather than only justifying the right one. All sums of money are written in words, all names and firms are invented, and nothing is drawn from any official publication.

• Find a rate of change from a table, from two points, or from a situation described in words, and know that the divisor is a difference rather than a coordinate
• Work back to the value at zero without mistaking the amount subtracted for the amount remaining
• Evaluate a function at a number and solve for the input that gives a required output
• Say what a coefficient and a constant mean in a real context, with no calculation
• Build a linear model from a description, attaching the recurring quantity to the variable, and test it at zero
• Recognise the specific slips that generate wrong answers: the inverted fraction, the flipped sign, the intermediate value taken as final

Written for this catalogue, with no source document. The content is the linear-function material named in the College Board's own public description of the Algebra domain — rate of change, initial value, linear functions and linear models in context; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test also asks some questions in a student-produced response format, which a multiple-choice bank cannot reproduce. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.