SAT Math — rewriting exponential models: period, rate and initial value

Among the hardest Advanced Math questions of the digital SAT are those that give an exponential model in one form and ask for an equivalent form that reveals something specific: the rate per month instead of per year, the factor per hour, the initial value, or which of two similar-looking models grows faster. The College Board's description of the domain asks students to rewrite expressions and to choose the form of a function that displays a given feature. This material trains exactly that.

You rewrite 500(1.21)^t as 500(1.1)^(2t), convert a yearly model to months with the exponent m/12, and read what the exponent 4t says about quarterly growth. You extract the daily percent lost from a half-life model, the yearly growth rate of a quantity that doubles every ten years, and the hourly factor of 200(4)^(t/3). Other questions split 3^(t + 2) to show the value at t = 0, rewrite 2^(3x) as 8^x, and find the base b that makes 100(1.44)^(t/2) = 100b^t.

Two questions target common reasoning errors: adding percent changes over two years instead of multiplying the factors, and assuming that 1.5 percent twice a year equals 3 percent once a year. A last question interprets the base of a model whose exponent is t/2, a form in which the factor applies once every two years rather than every year.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember. Approximate rates are checked with a calculator, as the digital SAT allows throughout the Math section.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

  • Rewrite exponential expressions with the power rules to change the period of growth
  • Convert a model between time units, such as years and months
  • Find the growth or decay rate per unit of time hidden in an equivalent form
  • Choose the equivalent form that shows the initial value or the rate
  • Compare percent changes over several periods by multiplying factors

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: equivalent expressions and nonlinear functions — exponential expressions and models), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.

Sample question

The value, in dollars, of an account $t$ years after it was opened is modeled by $V(t) = 2{,}000(1.02)^{4t}$. Which statement is true about the model?

See the answer

The value grows by 2 percent every 3 months.

The exponent $4t$ counts quarter-years: in each quarter (3 months) the value is multiplied by $1.02$, a 2 percent increase. Over a year the factor is $1.02^4 \approx 1.0824$, about 8.24 percent, so exactly 8 percent is false; 2 percent per year ignores the 4 in the exponent.

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