10.06 Modeling with functions (M)
Introduction
The wide variety of functions we've learned so far can be used to model and understand the world we live in. In this lesson, we will focus on how to choose a model that fits a given situation and combine models to help us understand and interpret complex scenarios. Recall the modeling process:
Each time we model a real-world situation we should:
Identify the essential features of the problem
Create a model using a diagram, graph, table, equation or expression, or statistical representation
Analyze and use the model to find solutions
Interpret the results in the context of the problem
Verify that the model works as intended and improve the model as needed
Report on our findings and the reasoning behind them
Identify the problem and create a model
When choosing a model to represent and solve real-world problems, we can use what we know about the key features of the functions we've learned so far and match them to the behavior of the problem.
- Linear functions: increase or decrease by a constant value, have infinite domain and range, can have both x and y-intercepts. Best used to model constant changes.
- Absolute value functions: have a constant rate of change, increase and decrease on different intervals, are symmetric, have infinite domain and restricted range. Best used to model deviations from a central value.
- Exponential functions: grow or decay by a constant factor, have an infinite domain but restricted range, do not have an x-intercept (never truly equal to zero). Best used to model percent changes.
- Quadratic functions: change direction, gradually changing rate of change, minimum or maximum values, can have more than one x-intercept. Best used to model projectile motion or situations that change directions smoothly.
- Piecewise functions: can have different characteristics at different intervals. Best used to model situations that change differently at different times.
Examples
Example 1
Drop tower rides are designed to lift passengers and then drop them in a freefall unexpectedly. Design a drop tower ride and create a mathematical model that describes the ride.
The behavior of a real-world situation can help us choose a mathematical model to represent it. Match the characteristics of each of the functions we've learned so far to the context to make the best decision.
Combining functions
Different functions can be used to model different situations. Sometimes, it takes a combination of functions to see the full picture. Some common function combinations:
- \text{Revenue}=\text{Price per unit}\cdot \text{Quantity sold}
- \text{Profit}=\text{Revenue} - \text{Cost}
- Newton's Law of Cooling: T(t)=T_s+(T_0-T_s)e^{-kt}
- Position of a falling object: d=\text{height}+ax^2
Examples
Example 2
To start a freeze-dried candy business, Carlee must purchase a freeze dryer and then supplies like candy, packaging, and labels for each product they sell.
Create a function that models Carlee's costs.
Determine a reasonable selling price and create a function to model Carlee's revenue.
Combine your functions from (a) and (b) to create a function that models the business owner's profit.
Use your cost, revenue, and profit models to write a business plan for the business owner.
Applying mathematics to real-world scenarios can involve using different functions to model different behaviors. In some cases, we need to combine multiple functions to get the full picture.