An exponential model's exponent counts the growth or decay periods

A growth or decay factor applies once per stated time interval

In $P(t)=500(0.8)^{t/3}$, the starting value is $500$. The factor $0.8$ means $80\%$ remains, or a $20\%$ decrease, every three years.

After six years, the exponent is $2$, so $P(6)=500(0.8)^2=320$. The overall decrease is $36\%$, not $40\%$. Each decay acts on the remaining amount.

$\displaystyle 500\cdot \left(0.8\right)^{\frac{6}{3}}$

Evaluate after two three-year intervals.

$\displaystyle \left(0.8\right)^{\frac{1}{3}}$

The equivalent one-year multiplier.

Equal differences suggest a linear model; equal ratios suggest an exponential one

Equal differences in the outputs suggest a linear model; equal ratios between positive outputs suggest an exponential model when the inputs are equally spaced. A quadratic has constant second differences at equally spaced inputs.

Enter the paired values in a table, choose the model the question specifies, and interpret its coefficients. A factor of $1.08$ means an $8\%$ increase per input unit, while $0.92$ means an $8\%$ decrease.

Distinguish an exact model from a measured trend

For measured data, a regression estimates a relationship. A prediction between the observed inputs is interpolation; a prediction beyond them is extrapolation and may be less reliable.

The SAT version automatically checks log mode for applicable regressions, including the exponential model used here. Do not assume a fitted curve passes through every observation, or that a high fit statistic proves causation. For coefficient recovery from an exact stated model, verify the fitted expression against the original conditions.

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