Concept 1

Decide whether a quadratic or exponential model fits a context or dataset.

Choosing between quadratic and exponential models based on how data changes.

Core Idea

If successive outputs change by a constant difference of differences (second differences are equal), the data is quadratic. If successive outputs change by a constant ratio, the data is exponential.

Understanding

When the SAT gives you a table or scenario and asks which model fits, focus on how the outputs change:

  • Compute first differences (the change between consecutive outputs). If those first differences are constant, the relationship is linear — not what this question is asking.

  • If the first differences themselves change by a steady amount, the pattern is quadratic. Example: outputs 2, 5, 10, 17 have first differences 3, 5, 7 and second differences 2, 2.

  • If each output is a fixed multiple of the previous one, the pattern is exponential. Example: outputs 3, 6, 12, 24 each double.

Context clues help too. Projectile motion, area problems, and revenue optimization typically involve quadratics. Population growth, radioactive decay, and compound interest typically involve exponentials.

Step by Step

  1. List the output values for equally spaced inputs.
  2. Compute first differences between consecutive outputs.
  3. If first differences are constant → linear (rule it out).
  4. Compute second differences. If constant → quadratic.
  5. Instead compute ratios of consecutive outputs. If constant → exponential.

Misconceptions

  • Confusing 'increases by a fixed amount each step' (linear) with 'increases by a fixed percentage each step' (exponential).
  • Assuming any curved graph must be exponential — parabolas are curved too.
  • Forgetting to check that inputs are equally spaced before comparing differences or ratios.
Question

Worked Example

A researcher records the number of bacteria in a sample every hour. The counts are 200, 600, 1800, and 5400 at hours 0, 1, 2, and 3, respectively. Which type of function best models the number of bacteria as a function of time?

Select an answer to see the explanation