AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/1/1-13)
Unit 1 · Topic 1.13
1.13 Function Model Selection and Assumption Articulation
To model data or a situation, you choose the type of function whose pattern of change matches. Then you state the assumptions behind the model and the inputs and outputs that make sense. This topic is about choosing and justifying, not yet about building the equation.
Key terms
- mathematical model
- linear model
- quadratic model
- piecewise function
- assumption
Matching patterns to function types
Look at how the output changes over equal steps of the input:
- Linear: roughly constant rate of change (first differences about the same).
- Quadratic: roughly linear rates of change (second differences about the same), or data that is roughly symmetric with one maximum or one minimum.
- Polynomial of degree n: roughly constant nonzero nth differences, or a pattern with several real zeros or several turning points.
- Piecewise: the data or situation behaves differently on different intervals.
Clues from geometry
Area involves two dimensions multiplied together, so area contexts often lead to quadratic models. Volume involves three, so volume contexts often lead to cubic models.
For example, cutting squares of side x from the corners of a rectangular sheet and folding up the sides gives a box with volume x(length − 2x)(width − 2x), a cubic.
How many points pin down a polynomial
Any n + 1 points with different inputs can be fit exactly by a polynomial of degree n or less. Two points determine a line, three points a quadratic (or less), four points a cubic (or less).
That doesn't make a high-degree model a good choice. A model should match the pattern and the context, not just pass through every point.
Piecewise-defined functions
A piecewise-defined function uses different formulas on intervals of the domain that don't overlap. It fits situations with clear phases, like a tank that fills at one rate, then drains at another, or a phone plan that charges a flat fee up to a limit and then charges per gigabyte.
Assumptions and restrictions
Every model rests on assumptions. Some are about what stays the same (the pump runs at a constant rate). Some are about how quantities change together (sales grow by a steady amount each week).
A model may need a restricted domain because of math (you can't divide by zero), the context (time can't be negative; a length must be positive), or the data (don't trust predictions far outside the inputs you have).
A model may need a restricted range too, such as rounding to whole numbers when counting people or cars.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Choosing a model from a table
A rocket's height (in meters) at times t = 0, 1, 2, 3, 4, 5 seconds is 3.0, 7.9, 10.9, 12.1, 11.0, 8.1. Which type of function is most appropriate: linear, quadratic or cubic? Justify.
Show the solutionHide the solution
- Step 1: The inputs are equally spaced. First differences: 4.9, 3.0, 1.2, −1.1, −2.9. They are not close to constant, so linear doesn't fit.
- Step 2: Second differences: −1.9, −1.8, −2.3, −1.8. These are roughly constant.
- Step 3: Roughly constant second differences mean the rates of change are changing at a roughly constant rate, which is the quadratic pattern. The data also rise to a single maximum and fall, which a parabola does.
Answer: Quadratic, because the average rates of change over equal intervals decrease by roughly the same amount each second (second differences near −2).
- Example 2Calculator allowed
A volume model and its domain
Squares of side x cm are cut from each corner of a 20 cm by 30 cm sheet, and the sides are folded up to make an open box. Write a model for the volume, state a sensible domain, and use a calculator to find the largest possible volume.
Show the solutionHide the solution
- Step 1: After cutting, the base is (30 − 2x) by (20 − 2x) and the height is x. So V(x) = x(20 − 2x)(30 − 2x), a cubic, as expected for a volume.
- Step 2: Every dimension must be positive: x > 0 and 20 − 2x > 0, so x < 10. The domain is 0 < x < 10.
- Step 3: Graph V on 0 < x < 10 and use the calculator's maximum feature.
Answer: V(x) = x(20 − 2x)(30 − 2x) for 0 < x < 10. The maximum volume is about 1056.306 cm³, when x ≈ 3.924 cm.
- Example 3
A piecewise model with assumptions
A tank holds 20 gallons. A pump adds water at 4 gallons per minute for 10 minutes, then shuts off, and the tank drains at 2 gallons per minute until it's empty. Write a model for the water W(t) after t minutes and name one assumption.
Show the solutionHide the solution
- Step 1: Phase 1 (0 ≤ t ≤ 10): start at 20 and add 4 per minute, so W(t) = 20 + 4t. At t = 10, W = 60.
- Step 2: Phase 2: start at 60 and lose 2 per minute after t = 10, so W(t) = 60 − 2(t − 10).
- Step 3: The tank is empty when 60 − 2(t − 10) = 0, so t − 10 = 30 and t = 40. Phase 2 runs for 10 < t ≤ 40.
- Step 4: The model assumes both rates stay exactly constant, and that the tank doesn't drain while the pump is running.
Answer: W(t) = 20 + 4t for 0 ≤ t ≤ 10, and W(t) = 60 − 2(t − 10) for 10 < t ≤ 40. One assumption: the pumping and draining rates are constant.
Common mistakes
- Choosing a model only because it passes through all the points. A degree-5 polynomial can hit six points exactly and still be a terrible model.
- Giving a domain that ignores the context, such as negative times or lengths, or extrapolating far beyond the data.
- Justifying a model choice with “it looks like it” instead of naming the pattern in the rates of change.
On the exam
- Free-response modeling questions often ask you to state an assumption or a limitation of the model. Make it specific to the context, such as “the price per item stays constant.”
- When asked which model is appropriate, support the choice with a pattern in the data: constant differences, constant second differences, or a single maximum.
Connected topics
Videos
Check yourself
4 questions on 1.13 Function Model Selection and Assumption Articulation. Pick an answer to see if you got it, and why.
The number of subscribers to a new streaming service is modeled by S(t) = 1200 + 85t, where t is the number of months since the service launched. Which of the following is an assumption of this model?
The volume of water in a reservoir, in millions of gallons, was recorded each month for 10 months. A cubic regression V(t) fits the data well for 0 ≤ t ≤ 10, where t is in months. The model gives V(14) = −35. Which of the following is the best conclusion?
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| h(x) | 4 | 4 | 10 | 34 | 88 | 184 |
Table of values
What is the least possible degree of h?
An open-top box is made from a rectangular sheet of cardboard that is 20 inches by 30 inches. A square with side length x inches is cut from each corner, and the sides are folded up. The volume of the box, in cubic inches, is modeled by V(x) = x(20 − 2x)(30 − 2x).
Invented scenario
Which of the following is the most appropriate domain for V in this context?
0 of 4 answered