Unit 1 · Topic 1.14 · about 30 minutes · 7 parts · 12 scored checks, 1 of them multiple choice

Function Model Construction and Application

Built from College Board's course framework · Every answer key independently checked

Graph of a fountain's water arc, h(x) = -1/3 (x - 2) squared + 3, from the nozzle at (0, 5/3) up to its highest point at (2, 3) and down to the ground at (5, 0). Horizontal distance in meters is on the x-axis and height in meters on the y-axis.

Part 1 Think first

Twice as far from the speaker

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

A sound meter 2 meters from a speaker reads an intensity of 40 units. You step back to 4 meters, twice as far. About what does the meter read now?

In short. Building a model means turning what you know about a situation into a specific function, then using it to answer questions with the right units.

A model can be built from restrictions in the context, from transformations of a parent function, or with technology and a linear, quadratic, cubic or quartic regression.
A piecewise model combines these techniques on different intervals, and quantities that are inversely proportional are often modeled by rational functions.
A model answers questions about values, rates of change, average rates of change and changing rates, and every answer carries units from the context.

Part 2 Learn it

Four short ideas, each checked as you go

2a. Build it from what you know

Often the context hands you the features of a model, and a parent function plus transformations (1.12) does the rest.

A fountain's arc of water rises to a highest point 3 meters up, 2 meters out from the nozzle, and lands 5 meters out. A quadratic fits the single symmetric maximum, and the vertex gives the transformation:

h (x) = a (x - 2)^{2} + 3

The landing point is a restriction the model must meet: $h (5) = 0$ . So $0 = a \cdot 9 + 3$ , and $a = - \frac{1}{3}$ . Context also restricts the domain: $0 \leq x \leq 5$ , from the nozzle to the ground.

Quick check · Calculate

No calculator. A ball's path is modeled by $h (x) = a (x - 4)^{2} + 6$ , where $x$ and $h$ are in meters. It reaches its highest point at $(4, 6)$ and leaves the thrower's hand at $(0, 2)$ . What is $a$ ?

Your answer

2b. Build it with a regression

When a model comes from data, technology can build it. A graphing calculator fits linear, quadratic, cubic and quartic regressions: the polynomial of that degree that comes closest to the data.

Your job is to choose which one to run, using the clues from 1.13: constant differences, a single symmetric peak, area or volume, several turns. The calculator then gives coefficients, which you round sensibly and use.

On the AP Exam a graphing calculator is required for some questions, in multiple-choice Part B and free-response Part A. A regression question would be one of those.

Quick check · Decide and justify

No calculator. A packaging company cuts a square of side $x$ centimeters from each corner of a sheet of cardboard and folds up the sides to make an open box. It measures the volume at $x = 1, 2, 3, 4, 5$ . The third differences of the volumes are constant and nonzero. Which regression should it run?

Which reason supports your answer?

2c. Build it in pieces

A piecewise-defined model can combine these techniques, one per interval. A rideshare driver's earnings on a shift might be a flat guarantee for the first two hours and a linear rate after that; a car's speed might grow linearly while it accelerates and then stay constant on the highway.

Build each piece from what happens on its interval, and check that the pieces meet where they should. For a car that speeds up from 0 at 3 meters per second each second for 10 seconds, then holds its speed:

$v (t) = 3 t$ for $0 \leq t \leq 10$
$v (t) = 30$ for $t > 10$

Both pieces give 30 at $t = 10$ , so the speed has no jump.

Quick check · Fill the statement

No calculator. A bike rental costs 12 dollars for up to 2 hours, then 5 dollars for each extra hour. Tap a blank, then tap a phrase.

The cost model has for $0 < h \leq 2$ and for $h > 2$ .

2d. Inverse proportion and rational models

Two quantities are inversely proportional when one is a constant divided by the other, $y = \frac{k}{x}$ : doubling $x$ halves $y$ . Such data and situations are often modeled by rational functions.

Some quantities are inversely proportional to the square of a distance. The magnitudes of gravitational force and of electromagnetic force between two objects both follow

F (d) = \frac{k}{d^{2}}

so doubling the distance divides the force by 4, and tripling it divides by 9. To build the model, find $k$ from one known pair: if $F = 50$ when $d = 2$ , then $k = 50 \cdot 2^{2} = 200$ .

Quick check · Calculate

No calculator. The intensity of light from a bulb is inversely proportional to the square of the distance from it. A meter 1 meter away reads 36 units. What does it read 3 meters away?

Your answer units

Part 3 Watch it done

How high is the fountain's nozzle?

Watch each step, then do the last one yourself.

A fountain's arc of water reaches its highest point, 3 meters, at 2 meters out from the nozzle, and lands 5 meters out. The nozzle is above the ground. Build a model and find the nozzle's height.

Step 1 · Choose the parent and transform it
One symmetric maximum suggests a quadratic. With vertex $(2, 3)$ : $h (x) = a (x - 2)^{2} + 3$ , where $x$ is meters out from the nozzle and $h$ is meters above the ground.
Step 2 · Use the restriction
The water lands at $(5, 0)$ : $0 = a (5 - 2)^{2} + 3 = 9 a + 3$ , so $a = - \frac{1}{3}$ . The model is $h (x) = - \frac{1}{3} (x - 2)^{2} + 3$ for $0 \leq x \leq 5$ .
Why this step?
Why is $x = 0$ the nozzle, and not where the water lands?
The model measures $x$ as distance out from the nozzle, so the nozzle itself is at $x = 0$ . The domain starts there and ends where the water reaches the ground, at $x = 5$ .
Your step · Calculate

No calculator. Finish it. How high is the nozzle, in meters? Give an exact fraction or a decimal to three places.

Your answer meters

Part 4 Try it

A regression to predict visitors

A museum counts weekly visitors, in hundreds, $w$ weeks after a new exhibit opens: 5.0, 16.2, 25.8, 33.8 and 40.2 for $w = 0$ to 4. The first differences, 11.2, 9.6, 8.0 and 6.4, fall by 1.6 each week, so a quadratic regression is the right one to run.

Weekly visitors, in hundreds
$w$ (weeks)	$V (w)$
0	5.0
1	16.2
2	25.8
3	33.8
4	40.2

Try it · Calculate

Graphing calculator required. A museum counts weekly visitors, in hundreds, $w$ weeks after a new exhibit opens: 5.0, 16.2, 25.8, 33.8 and 40.2 for $w = 0, 1, 2, 3, 4$ . Run a quadratic regression on these data, and use it to predict the number of visitors, in hundreds, in week 6.

Your answer hundred visitors

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page.

Free response is 37.5% of the exam score. This one is shaped like Question 2, Modeling a Non-Periodic Context.

Free-response practice · Question 2 style

Graphing calculator required. The brightness of a streetlight on the ground, $I (d)$ in lux, is inversely proportional to the square of the distance $d$ in meters from the bulb. At 2 meters the brightness is 180 lux. Build a model for $I$ . Then find the average rate of change of $I$ from $d = 2$ to $d = 6$ , with units, and interpret it in context.

Your answer stays on this page. It is never sent or saved.

Model answer

Inverse square: $I (d) = \frac{k}{d^{2}}$ . From $180 = \frac{k}{4}$ , $k = 720$ , so $I (d) = \frac{720}{d^{2}}$ . Then $I (6) = \frac{720}{36} = 20$ lux. The average rate of change from 2 to 6 meters is $\frac{20 - 180}{6 - 2} = - 40$ lux per meter. On average, the brightness drops 40 lux for each meter farther from the bulb between 2 and 6 meters, though it drops faster near the bulb than farther away.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes $I (d) = \frac{k}{d^{2}}$ and finds $k = 720$ .
Finds $I (6) = 20$ lux.
Computes the average rate of change as $- 40$ lux per meter, with units.
Interprets it: on average, brightness falls 40 lux per meter between 2 and 6 meters.

Part 6 Check yourself

Filling a tank

The two case questions use the same data. Answer them from the table.

A water tank is filled from a hose. After 10 minutes the flow is turned down. $V (t)$ is the volume of water in liters, $t$ minutes after the hose is turned on.

Water in the tank
$t$ (minutes)	$V (t)$ (liters)
0	0
5	150
10	300
15	350
20	400
25	450

Case question 1 · Multiple choice

No calculator. A piecewise model starts with $V (t) = 30 t$ for $0 \leq t \leq 10$ . Which second piece, for $10 < t \leq 25$ , fits the table?

Case question 2 · Calculate

No calculator. What is the average rate of change of $V$ from $t = 0$ to $t = 20$ , in liters per minute?

Your answer L/min

The rest stand on their own and do not use the case.

Check yourself · Sort it

No calculator. Tap a situation, then the model you would build.

Rational

Quadratic

Linear

Check yourself · Spot the error

No calculator. A student uses the gravity model $F (d) = \frac{800}{d^{2}}$ , with $F$ in newtons and $d$ in meters, to find the average rate of change from $d = 2$ to $d = 4$ . Tap the first line with a mistake.

Part 7 Keep it

Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 1.13 Function Model Selection and Assumption Articulation · Fill the statement

No calculator. Tap a blank, then tap a number.

Through any 4 points with distinct inputs there is a polynomial of degree or less, and through any 7 such points there is one of degree or less.

From 1.11 Equivalent Representations of Polynomial and Rational Expressions · Decide and justify

No calculator. A theater's ticket revenue, in hundreds of dollars, at a price of $x$ dollars is $R (x) = - 2 x^{2} + 40 x$ . Two equivalent forms are the factored form, $R (x) = - 2 x (x - 20)$ , and the vertex form, $R (x) = - 2 (x - 10)^{2} + 200$ . The manager asks what price gives the most revenue. Which form answers that directly?

Which reason supports your answer?

In three lines

Build a model from restrictions and transformations of a parent function, or from a regression chosen by the pattern in the data.
Piecewise models combine techniques on intervals; inversely proportional quantities suggest a rational model such as $\frac{k}{d^{2}}$ .
Use the model for values, rates and average rates, always with units from the context.

Unit 1 check: 14 questions from every lesson in Unit 1

Vocabulary

Function model: A function built to describe a data set or a situation and to answer questions about it.
Regression: A model built with technology that comes as close as possible to a data set, such as a linear, quadratic, cubic or quartic regression.
Inversely proportional: Related by $y = \frac{k}{x}$ , so that doubling one quantity halves the other.
Inverse square: A relationship $y = \frac{k}{x^{2}}$ , such as gravitational and electromagnetic force against distance.
Units of a rate: Output units per input unit, such as liters per minute or lux per meter.

Course alignment, for teachers

AP Precalculus topic 1.14, Unit 1: Polynomial and Rational Functions.

Skill 1.C: Construct new functions, using transformations, compositions, inverses, or regressions, that may be useful in modeling contexts, criteria, or data, with and without technology.
Skill 3.B: Apply numerical results in a given mathematical or applied context.

This lesson maps all 6 essential knowledge statements of the topic to the part that teaches each one: 6 are checked by a scored question. The map is in the lesson file.