Overview
This topic explores how various types of functions can be used to model real-world situations. You'll learn how to translate between verbal descriptions, algebraic equations, graphs, and tables to analyze patterns and solve applied problems.
Key Concepts and Applications
- Linear Models: Represent constant rate of change. Equation:
y = mx + b
- Quadratic Models: Represent parabolic motion, such as projectile paths. Equation:
y = ax^2 + bx + c
- Exponential Models: Represent growth/decay, population, interest, etc. Equation:
y = ab^x
- Logarithmic Models: Used in pH scale, Richter scale, and intensity models
- Piecewise Functions: Represent functions with different expressions over different intervals (e.g., tax brackets, delivery rates)
- Verbal to Algebraic: Translate descriptive scenarios into equations
- Graphical Interpretation: Use intercepts, slope, curvature, asymptotes, and continuity to analyze behavior
- Tabular Representation: Identify patterns or rates of change in tables of values
Step-by-Step Example
Problem: A car depreciates in value by 15% per year. If its initial value is $20,000, write a model and find its value after 3 years.
Step 1: Recognize it’s exponential decay:
V(t) = 20000(0.85)^t
Step 2: Substitute t = 3 into the equation:
V(3) = 20000(0.85)^3 ≈ 20000(0.614125) ≈ 12282.50
Answer: Approximately $12,282.50 after 3 years
Quick Tip
When modeling word problems, identify the type of function by looking for keywords: constant change (linear), percent change (exponential), maximum/minimum (quadratic), etc.