Linear versus Exponential Change

Objective: Predict a linear sequence from a constant difference.

Prerequisite skills: rates, tables, and powers. A linear relationship adds a fixed amount for every equal input step. Subtract consecutive outputs to test the difference; the initial value is independent of that change. Common error: multiplying by the difference instead of adding it. Restrictions and interpretation: Equal differences indicate linear change only for equal input intervals; equal ratios indicate exponential change.

Objective: Predict an exponential sequence from a constant ratio.

An exponential relationship multiplies by the same positive factor over equal input intervals. Its differences need not remain constant. The exponent counts how many multiplication intervals have elapsed. Common error: adding the growth factor instead of multiplying by it. Restrictions and interpretation: Equal differences indicate linear change only for equal input intervals; equal ratios indicate exponential change.

Objective: Compare linear and exponential outputs at a specified time.

Exponential models use a constant ratio while linear models use a constant difference. To compare them at a specific time, evaluate both with the same elapsed interval and subtract in the requested order. Common error: assuming exponential is larger at every early time. Restrictions and interpretation: Equal differences indicate linear change only for equal input intervals; equal ratios indicate exponential change. 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.