Linear vs. Exponential Models
Aligned to F-LE.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
A linear function adds the same amount each step (a constant difference); an exponential function multiplies by the same factor each step (a constant ratio). Use two points or a table to find the rule.
Worked examples
Example 1
A population doubles yearly, starting at 300. It multiplies by 2 each year, so P(x) = 300 × 2^x. After 3 years, P(3) = 300 × 2^3 = 2400.
Example 2
Halving is exponential too: a drug at 200 mg with half clearing each hour is A(t) = 200(0.5)^t.
Example 3
Decide by structure, not by shape. A table whose ratios are constant is exponential even if the first few values look linear.
Practice questions
- Write a linear function through (0, 4) and (1, 9).
- A car worth $20,000 loses half its value each year. Write the exponential function.
- A table shows 2, 6, 18, 54. Is it linear or exponential? Write its rule.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
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Modelos Lineales vs. Exponenciales
Una función lineal suma la misma cantidad en cada paso (una diferencia constante); una función exponencial multiplica por el mismo factor en cada paso (una razón constante). Usa dos puntos o una tabla para hallar la regla.
Ejemplo 1: Una población se duplica cada año, empezando en 300. Se multiplica por 2 cada año, así que P(x) = 300 × 2^x. Después de 3 años, P(3) = 300 × 2^3 = 2400.
Ejemplo 2: Reducir a la mitad también es exponencial: un medicamento de 200 mg del que se elimina la mitad cada hora es A(t) = 200(0.5)^t.
Ejemplo 3: Decide por estructura, no por forma. Una tabla cuyas razones son constantes es exponencial aunque los primeros valores se vean lineales.
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