Interpreting Initial Values and Growth Factors

Objective: Find the model value at time zero.

Prerequisite skills: exponential models and decimal percentages. In A(t)=A₀b^t, substitute t=0 to obtain A₀ because b⁰=1 for positive b. The base b is a factor per time unit, not an initial population or a percent by itself. Common error: treating the base as the initial amount. Restrictions and interpretation: The coefficient is the output at exponent zero; the rate depends on the unit represented by one exponent step.

Objective: Convert between an exponential factor and a percent change.

For a growth factor b>1, the percent growth is 100(b−1)%. For a decay factor 0<b<1, the percent decrease is 100(1−b)%. The factor includes the retained original 100%. Common error: calling a factor of 1.12 a 112% increase. Restrictions and interpretation: The coefficient is the output at exponent zero; the rate depends on the unit represented by one exponent step.

Objective: Interpret an exponent measured in several time units.

If A(t)=A₀b^(t/k), the factor b applies every k units of t because the exponent rises by one over that interval. The factor for one t-unit is b^(1/k), not b/k. Common error: dividing a growth factor by the interval length. Restrictions and interpretation: The coefficient is the output at exponent zero; the rate depends on the unit represented by one exponent step. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.