AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/1/1-14)
Unit 1 · Topic 1.14
1.14 Function Model Construction and Application
Once you've chosen a model type, you build the actual function, from given conditions, from transformations of a parent function, or from data with a calculator regression. Then you use it to predict values and rates in context, with units, and check that the answers make sense.
Key terms
- regression
- prediction
- piecewise-defined model
- contextual restrictions
Building a model from conditions
Sometimes the context gives you enough facts to write the function by hand. A parabola with a known vertex and one more point, a line through two points, or a polynomial with known zeros can all be written directly.
Transformations help here. Start with a parent function like x² or x³ and stretch, reflect and shift it to fit the conditions: a(x − h)² + k for a quadratic with vertex (h, k).
If the form is given with unknown constants, plug in known points to get a system of equations. For f(x) = ax² + bx + c through (0, 2), (1, 5) and (3, 5): c = 2, then a + b + 2 = 5 and 9a + 3b + 2 = 5. Solving gives a = −1 and b = 4, so f(x) = −x² + 4x + 2.
Regression with a calculator
With data, a graphing calculator can find the best-fitting linear, quadratic, cubic or quartic function. Enter the inputs and outputs as lists, choose the regression type that matches the pattern you identified, and store the equation.
On the exam, give regression coefficients to at least three decimal places, and use the stored, unrounded equation for later calculations so rounding errors don't pile up.
Rational models
When one quantity is inversely proportional to another, a rational function is natural. If y is inversely proportional to x, then y = k/x. If it is inversely proportional to the square of x, then y = k/x².
Gravitational force and electric force between two objects are both inversely proportional to the square of the distance between them. Double the distance and the force drops to 1/4; triple it and the force drops to 1/9.
Piecewise models
A piecewise model combines techniques. One piece might be linear and another quadratic or constant. Check that the pieces line up at the boundaries when the quantity can't jump.
Using and judging the model
Use the model to predict outputs, find inputs that give a target output, and compute average rates of change. Always attach units, and say what the number means in the context.
Ask whether the answer is reasonable. Predictions far outside the data (extrapolation) are risky, and a model can give impossible values, like a negative population, outside its sensible domain.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A parabola from conditions
An arch is 16 meters wide at the ground and 12 meters tall at its center. Put the left foot of the arch at (0, 0). Write a quadratic model for the height h(x) of the arch, and find the height 4 meters from the left foot.
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- Step 1: By symmetry, the top is halfway across, at (8, 12). Use vertex form: h(x) = a(x − 8)² + 12.
- Step 2: The right foot is at (16, 0). Use the left foot (0, 0): 0 = a(0 − 8)² + 12, so 64a = −12 and a = −3/16.
- Step 3: h(x) = −(3/16)(x − 8)² + 12, for 0 ≤ x ≤ 16.
- Step 4: h(4) = −(3/16)(16) + 12 = −3 + 12 = 9.
Answer: h(x) = −(3/16)(x − 8)² + 12; the arch is 9 meters tall 4 meters from the left foot.
- Example 2Calculator allowed
Cubic regression and an average rate
A food truck's weekly sales S, in hundreds of dollars, x weeks after opening are: (0, 12.0), (2, 19.5), (4, 21.1), (6, 20.3), (8, 22.6), (10, 31.9). Use a cubic regression to model S. Predict sales in week 7, and find the average rate of change of the model from x = 2 to x = 6.
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- Step 1: The sales rise, level off, then rise again: a pattern with a turning region, so a cubic is reasonable.
- Step 2: Cubic regression gives S(x) ≈ 0.094x³ − 1.354x² + 6.154x + 11.971 (more precisely 0.09375x³ − 1.35357x² + 6.15357x + 11.97143).
- Step 3: S(7) ≈ 20.878, which is about 2,088 dollars.
- Step 4: S(2) ≈ 19.614 and S(6) ≈ 20.414. Average rate = (20.414 − 19.614)/(6 − 2) ≈ 0.200 hundred dollars per week.
- Step 5: In context, from week 2 to week 6 the model's sales grow by about 20 dollars per week on average.
Answer: S(7) ≈ 20.878 hundred dollars (about 2,088 dollars); the average rate of change from x = 2 to x = 6 is about 0.200 hundred dollars (20 dollars) per week.
- Example 3
An inverse-square model
The brightness I of light from a small bulb is inversely proportional to the square of the distance d from the bulb. At 3 meters the brightness is 40 units. Write a model, and find the brightness at 6 meters.
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- Step 1: Inverse square means I = k/d².
- Step 2: Use the data: 40 = k/3², so k = 40 · 9 = 360.
- Step 3: I(d) = 360/d², for d > 0.
- Step 4: I(6) = 360/36 = 10. Doubling the distance cut the brightness to 1/4, as an inverse-square model should.
Answer: I(d) = 360/d²; at 6 meters the brightness is 10 units.
Common mistakes
- Rounding regression coefficients early and then using the rounded equation. Store the full equation and round only the final answer to three decimal places.
- Reporting a number without units or without saying what it means in the context.
- Trusting predictions far outside the data. A cubic model for 10 weeks of sales predicts about 5,290 dollars in week 12, which may not be realistic.
On the exam
- The non-periodic modeling free-response question asks you to build a model (often solving for constants from given points), use it to find values and average rates of change with units, and explain an assumption or limitation.
- When you use a regression, name the type (for example, “cubic regression”) and write the equation with its coefficients.
Connected topics
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Check yourself
4 questions on 1.14 Function Model Construction and Application. Pick an answer to see if you got it, and why.
| x (weeks) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| D (thousands) | 2.1 | 3.9 | 8.2 | 14.1 | 21.8 | 32.3 |
Invented data
A quadratic regression is used to model D as a function of x. Based on this model, which of the following is closest to the predicted number of downloads, in thousands, at x = 7 weeks?
The brightness B of a lamp, measured by a light meter, is inversely proportional to the square of the distance d, in meters, between the meter and the lamp. When d = 2, the brightness is 90 units. What is the brightness when d = 6?
A bike rental shop charges $4 for a rental of 1 hour or less. For a rental longer than 1 hour but no longer than 6 hours, it charges the $4 plus $2.50 for each hour after the first, billed by the minute, so part of an hour costs that fraction of $2.50. Any rental longer than 6 hours costs the day rate of $18. The cost C(h), in dollars, of a rental lasting h hours is modeled by a piecewise-defined function. What is C(4.5)?
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 closest to the maximum possible volume of the box?
0 of 4 answered