Modeling Real-World Problems with Functions
Aligned to F-IF.C.7 — Common Core State Standards for Mathematics.
What this lesson teaches
Choose a function type — linear, quadratic, exponential — that matches how a quantity changes, then use it to make predictions.
Worked examples
Example 1
A bacteria count doubling each hour is exponential: N = N₀·2^t. Starting at 100, after 3 hours N = 100·2³ = 800.
Example 2
A cooling cup of coffee starts 60°C above room temperature and loses 10% of that gap each minute. The model is T(t) = 20 + 60(0.9)^t with room temperature 20°C. After 10 minutes: 20 + 60(0.349) = 40.9°C. The gap shrinks fast at first and then slowly — the signature of exponential decay.
Example 3
Common mistake: fitting a linear model to data that bends. A population going 100, 200, 400, 800 gains 100, then 200, then 400 — the DIFFERENCES are not constant, so it is not linear. Check the differences for a linear fit and the RATIOS for an exponential one: here 200/100 = 400/200 = 2, so it is exponential. A straight line through that data will be badly wrong at both ends.
Practice questions
- Is steady $50/week savings linear or exponential?
- Model a population doubling every year starting at 5.
- Which function fits a value that grows by a fixed percent each period?
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.
▶ Watch this lessonEn español
Modelar problemas del mundo real con funciones
Elige el tipo de función — lineal, cuadrática, exponencial — que corresponda a cómo cambia una cantidad y úsala para hacer predicciones.
Ejemplo 1: Una cantidad de bacterias que se duplica cada hora es exponencial: N = N₀·2^t. Comenzando en 100, después de 3 horas N = 100·2³ = 800.
Ejemplo 2: Una taza de café que se enfría empieza 60°C por encima de la temperatura ambiente y pierde el 10% de esa diferencia cada minuto. El modelo es T(t) = 20 + 60(0.9)^t con una temperatura ambiente de 20°C. A los 10 minutos: 20 + 60(0.349) = 40.9°C. La diferencia se reduce rápido al principio y luego despacio — la firma del decaimiento exponencial.
Ejemplo 3: Error común: ajustar un modelo lineal a datos que se curvan. Una población que va 100, 200, 400, 800 gana 100, luego 200, luego 400 — las DIFERENCIAS no son constantes, así que no es lineal. Revisa las diferencias para un ajuste lineal y las RAZONES para uno exponencial: aquí 200/100 = 400/200 = 2, así que es exponencial. Una recta trazada sobre esos datos fallará mucho en ambos extremos.
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