SAT Math — exponential models and nonlinear relationships

There is one distinction underneath almost every exponential question on the test, and a learner who has it secure can answer most of them by reading rather than calculating. A quantity that gains the same AMOUNT each period is linear. A quantity that gains the same PROPORTION of itself each period is exponential. The test is not how fast something changes but whether the change is measured against the quantity itself, and that is why five hundred dollars of depreciation a year is a straight line while three percent interest is not.

Three questions here give a situation in words and ask for the equation. Each carries, among its wrong options, the straight line a careless reader builds by adding the percentage as though it were an amount — and each explanation evaluates both models at one concrete time so the difference is shown rather than asserted. Another wrong option in each uses the percentage itself as the multiplier, which turns a five percent rise into the loss of nineteen twentieths of the town.

Two questions read an equation the other way. One asks what the coefficient means, answered by setting the time to zero so the power becomes one and the starting value is all that remains. One asks what a multiplier below one means: it is the share that SURVIVES each period, so a multiplier of ninety-two hundredths is an eight percent loss, not a ninety-two percent one.

Two evaluate a model at a stated time, and the second is the one worth slowing down for. Its exponent divides the elapsed time by the length of one period, because the exponent counts periods and not days. A learner who substitutes the number of days directly gets an answer that is wrong by a factor of a thousand and looks perfectly reasonable.

Two ask which of four described situations fits a named model, with the other three all belonging to the opposite family — including a quantity that doubles, which is exponential even though no percentage is mentioned anywhere.

The last three leave exponentials for other nonlinear relationships: evaluating one where the square binds to the variable and not to the coefficient, solving one backwards through a fractional coefficient, and counting where a line meets a curve. That final question turns on a habit worth keeping: setting the two sides equal and dividing through by the variable loses a solution silently, because dividing by a variable assumes it is not zero, and here it is.

Every firm and town is invented, every sum of money written in words, and nothing is drawn from any official publication.

• Tell an exponential situation from a linear one by asking whether the change is a fixed amount or a fixed share
• Turn a percentage rise or fall into the right multiplier, and never use the percentage itself
• Read the starting value out of a model by setting the time to zero
• Read a multiplier below one as the share that survives, and subtract from one to get the loss
• Count periods rather than units of time when the exponent has a divisor in it
• Evaluate a nonlinear relationship with the exponent bound to the variable, and solve one backwards
• Find where a line meets a curve without dividing through by a variable and losing a solution

Written for this catalogue, with no source document. The content is the exponential and nonlinear material named in the College Board's own public description of the Advanced Math domain — exponential functions, nonlinear relationships in one and two variables, and systems combining a linear with a nonlinear equation; 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.