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Problem-Solving and Data AnalysisMediumChoosing between a linear and an exponential model

Problem-Solving and Data Analysis practice question

The table below shows the value of y at four consecutive values of x. x = 0, y = 3 x = 1, y = 6 x = 2, y = 12 x = 3, y = 24

Which type of model best fits these data, and why?

  1. A. An exponential model, because each y-value is a constant multiple of the previous one.Correct
  2. B. A linear model, because each y-value increases by a constant amount.
  3. C. A linear model, because the x-values increase by a constant amount.
  4. D. Neither model fits, because the y-values are not all multiples of 3.

Answer: A. An exponential model, because each y-value is a constant multiple of the previous one.

Check the differences first: 6 - 3 = 3, 12 - 6 = 6, 24 - 12 = 12. These are not constant, so the data are not linear. Now check the ratios: 6/3 = 2, 12/6 = 2, 24/12 = 2. A constant ratio between successive y-values is the signature of exponential growth.

Why the other answers are wrong

B. A linear model, because each y-value increases by a constant amount.
The y-values do increase, but not by a constant amount — the increases are 3, then 6, then 12. Growing by a constant amount is what makes data linear.
C. A linear model, because the x-values increase by a constant amount.
Evenly spaced x-values are what let you compare differences and ratios in the first place; they say nothing about which model fits. It is the pattern in y that decides.
D. Neither model fits, because the y-values are not all multiples of 3.
Every y-value here is in fact a multiple of 3, and in any case a model is chosen from the pattern of change, not from whether values share a factor.

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