Function Model Selection and Construction

🧭 Function Model Selection and Construction

Part 1 of 7 — Choosing a Function Model

Topic Overview

What a Model Is

A function model is a function chosen to describe how one quantity in a real situation depends on another. The model is never the situation itself: it captures the pattern in the data or the geometry, and we use it to predict, to interpret and to compare.

The first decision is always the type of function. Each type matches a different pattern in how the output changes:

Key Fact: On the AP exam you are asked to justify the type you choose. The justification names the pattern: "the first differences are roughly constant, so a linear model is appropriate."

Successive Differences on a Table

When the inputs in a table are equally spaced, subtract consecutive outputs.

A linear pattern

First differences: $4, 4, 4, 4$. The output rises by the same amount for every unit of input, so the rate of change is constant and a linear model fits.

A quadratic pattern

First differences: $2, 4, 6, 8$. These are not constant, so the model is not linear. But the first differences themselves grow by $2$ each time:

Second differences: $2, 2, 2$.

Constant second differences mean the rate of change changes at a constant rate: the graph is concave up and a quadratic model fits.

In general

With real data the differences are only roughly constant. That is fine: say "roughly constant" in your justification.

Key Fact: When two candidate models are drawn over the same data, the better one is the one whose pattern of change matches the data, not the one with the simpler formula. A line has one constant rate of change and cannot bend; if its errors follow a pattern (below the data at both ends, above in the middle), the data are curved.

Reading a context instead of a table

• A car on cruise control: distance grows by the same amount each minute → linear.
• The height of a thrown ball: rises, reaches one maximum, falls symmetrically → quadratic.
• The volume of a box whose dimensions each depend on $x$: a product of three lengths → cubic polynomial.
• An electricity bill that charges one rate for the first $500$ kWh and a higher rate after that → piecewise.
• The time a trip takes at speed $v$ over a fixed distance $d$: $t = \frac{d}{v}$, inversely proportional → rational.

Two candidate models. 🎯

The eight data points above have first differences $1.1, 1.7, 2.2, 2.7, 3.4, 3.9, 4.7$. Model A is a line; Model B is a parabola.

Choose the model type. 🎯

Work the differences. 🔽

A linear table. ✍️

1) The common first difference (the change in $y$ for each step of $2$ in $x$):

2) The rate of change of $y$ per unit of $x$:

3) The value of $y$ the linear model predicts at $x = 8$:

Key Takeaways — Part 1

• A model is chosen for its pattern of change, and the justification names that pattern
• Constant first differences → linear; constant second differences → quadratic; constant $n$th differences → degree $n$
• Several zeros or extrema, or a product of lengths → polynomial; different rules on different intervals → piecewise; inverse proportion → rational
• Real data give roughly constant differences: say so

Up Next: Part 2 — building linear and quadratic models from points, and letting the calculator find a regression.

🧭 Function Model Selection and Construction

Part 2 of 7 — Linear and Quadratic Models from Data

A Linear Model from Two Points

A linear model $y = mx + b$ has two parameters, and each has a meaning in context:

Example. A plant is $8$ cm tall when it is planted ($t = 0$ days) and $20$ cm tall on day $6$.

$m = \frac{20 - 8}{6 - 0} = \frac{12}{6} = 2 \text{ cm per day}, \qquad b = 8 \text{ cm}$

$h(t) = 8 + 2t$

In words: the plant grows $2$ centimeters per day, starting from $8$ centimeters.

Example without a starting value. A taxi ride of $3$ miles costs $50$ units and a ride of $8$ miles costs $85$ units. The slope is $\frac{85 - 50}{8 - 3} = \frac{35}{5} = 7$ units per mile. Using the point $(3, 50)$:

$50 = 7(3) + b \quad \Rightarrow \quad b = 29, \qquad C(x) = 7x + 29$

The intercept $29$ is the cost before any miles are driven: the base fare.

Key Fact: Always attach units to a rate: "$7$ units per mile," not just "$7$."

A Quadratic Model from Its Features

From the vertex

If you know the maximum or minimum point $(h, k)$, start from vertex form $y = a(x - h)^2 + k$ and use one more point to find $a$.

Example. A ball is launched from the ground ($h(0) = 0$) and reaches its maximum height of $36$ feet at $t = 1.5$ seconds.

$h(t) = a(t - 1.5)^2 + 36$

$0 = a(0 - 1.5)^2 + 36 = 2.25a + 36 \quad \Rightarrow \quad a = -16$

$h(t) = -16(t - 1.5)^2 + 36$

The negative $a$ confirms the graph opens downward: the ball rises and then falls.

From three points

Three points determine a quadratic $y = ax^2 + bx + c$. With $(0, 5)$, $(1, 8)$ and $(2, 15)$:

• $(0, 5)$: $c = 5$
• $(1, 8)$: $a + b + 5 = 8$, so $a + b = 3$
• $(2, 15)$: $4a + 2b + 5 = 15$, so $2a + b = 5$

Subtracting the second equation from the third: $a = 2$, then $b = 1$.

$y = 2x^2 + x + 5$

Check: at $x = 2$, $2(4) + 2 + 5 = 15$. ✓

Interpreting the parameters

In $y = ax^2 + bx + c$ the constant $c$ is still the output at input $0$. The sign of $a$ tells you concave up ($a > 0$, a minimum) or concave down ($a < 0$, a maximum). The vertex is at $x = -\frac{b}{2a}$.

The Calculator Framing: Regression

Real data never sit exactly on a line or a parabola. On the exam, Part B of the multiple-choice section and Part A of the free-response section allow a graphing calculator, and the expected tool is regression: the calculator finds the linear, quadratic or cubic function that best fits the data.

What you must write

1. State the model type and why (the pattern from Part 1).
2. State the regression equation, with coefficients rounded to three decimal places.
3. Keep full precision inside the calculator for every later computation. Round only the final answer.

Worked Example

The height, in centimeters, of a seedling $t$ weeks after planting:

First differences: $2.1, 1.7, 2.4, 1.7, 2.2$, roughly constant at about $2$. A linear model is appropriate.

The calculator's linear regression gives

$H(t) = 2.009t + 1.995$

Interpretation. The seedling grows at a rate of approximately $2.009$ centimeters per week, and the model gives an initial height of approximately $1.995$ cm.

Prediction. At $t = 8$ weeks, the stored full-precision model gives $H(8) \approx 18.064$ cm. Using the rounded coefficients instead gives $2.009(8) + 1.995 = 18.067$: the third decimal is already different. Use the stored model.

Key Fact: Write "approximately" whenever a value comes from a model or a rounded regression: "the model predicts the seedling will be approximately $18.064$ cm tall after $8$ weeks."

Build and interpret. 🎯

Use a regression model. ✍️

A calculator gives the quadratic regression $h(t) = -4.912t^2 + 19.835t + 1.502$ for the height of a ball in meters $t$ seconds after it is thrown. Use three decimal places.

1) The height, in meters, at the moment the ball is thrown:

2) The height, in meters, the model predicts at $t = 2$:

3) The average rate of change of height over the interval $[0, 2]$, in meters per second:

Exam habits for regression models. 🔽

Key Takeaways — Part 2

• Linear: slope is the rate of change with units, intercept is the starting value
• Quadratic: use vertex form when the maximum or minimum is known, or three points to solve for $a$, $b$, $c$
• On calculator sections, the model comes from regression: name the type, state the equation to three decimal places, keep full precision for later steps
• Say "approximately" for any value a model produces

Up Next: Part 3 — cubic and quartic models from geometry, with domains that come from the context.

🧭 Function Model Selection and Construction

Part 3 of 7 — Polynomial Models in Context

Building a Polynomial from Geometry

An open-top box is made from a $20$ cm by $16$ cm sheet of cardboard by cutting a square of side $x$ cm from each corner and folding up the sides.

$V(x) = x(20 - 2x)(16 - 2x)$

Expanding: $(20 - 2x)(16 - 2x) = 4x^2 - 72x + 320$, so

$V(x) = 4x^3 - 72x^2 + 320x,$

a cubic polynomial with leading coefficient $4$. A product of three lengths that each depend on $x$ is always degree $3$.

Domain in Context

Algebraically $V(x)$ is defined for every real number. In context, every dimension must be positive:

$x > 0, \qquad 20 - 2x > 0 \Rightarrow x < 10, \qquad 16 - 2x > 0 \Rightarrow x < 8$

The tightest restriction wins: the domain in context is $0 < x < 8$.

What the Zeros Mean

$V(x) = 0$ when $x = 0$, $x = 8$ or $x = 10$.

• $x = 0$: no square is cut, so there is no height and no box.
• $x = 8$: the two $8$ cm cuts use up the whole $16$ cm width, so the width is $0$.
• $x = 10$: outside the domain in context; it has no meaning for this box.

Key Fact: The zeros of a model are the inputs where the output quantity is zero. Always check whether a zero lies inside the domain in context before interpreting it.

Reading the Model on Its Domain

The graph shows $V(x)$ only for $0 < x < 8$, with open circles at both ends because neither endpoint produces a box.

The maximum, from the calculator

Using the calculator's maximum feature on the graph, the local maximum is at approximately $x = 2.945$, where $V \approx 420.110$. In context: cutting squares of side approximately $2.945$ cm gives the largest possible volume, approximately $420.110$ cubic centimeters.

Average rate of change, with units

$V(1) = 1 \cdot 18 \cdot 14 = 252, \qquad V(2) = 2 \cdot 16 \cdot 12 = 384$

$\text{Average rate of change on } [1, 2] = \frac{384 - 252}{2 - 1} = 132 \text{ cubic centimeters per centimeter}$

In words: as the cut grows from $1$ cm to $2$ cm, the volume increases at an average rate of $132$ cubic centimeters per centimeter of cut.

$V(3) = 3 \cdot 14 \cdot 10 = 420, \qquad V(4) = 4 \cdot 12 \cdot 8 = 384, \qquad \text{so on } [3, 4]: \frac{384 - 420}{1} = -36$

The negative sign says the volume is decreasing on that interval: past the maximum, bigger cuts make a smaller box.

Key Fact: A rate of change is a number with units and an interval: "cubic centimeters per centimeter, on $[3, 4]$."

Use the graph and the model $V(x) = x(20 - 2x)(16 - 2x)$. 🎯

A closed box. ✍️

A box has a square base of side $x$ cm and height $12 - x$ cm, so its volume is $V(x) = x^2(12 - x)$ cubic centimeters.

1) $V(4)$, in cubic centimeters:

2) The domain in context is $0 < x < c$. The value of $c$:

3) The average rate of change of $V$ on $[2, 4]$, in cubic centimeters per centimeter:

Say it the AP way. 🔽 For the open-top box, the calculator gives a maximum of $V \approx 420.110$ at $x \approx 2.945$.

Key Takeaways — Part 3

• A product of lengths that each depend on $x$ gives a polynomial whose degree is the number of factors
• The domain in context comes from requiring every quantity to make sense (positive lengths)
• Zeros are inputs where the output is zero; interpret only those inside the domain in context
• Find a maximum with the calculator; report it as "approximately … when $x$ is approximately …"
• An average rate of change is $\frac{\Delta \text{output}}{\Delta \text{input}}$ with units and an interval

Up Next: Part 4 — piecewise-defined functions: reading the brace and choosing the right rule.

🧭 Function Model Selection and Construction

Part 4 of 7 — Piecewise-Defined Functions: Notation and Evaluation

One Function, Several Rules

Most functions use one rule for every input, like $f(x) = 2x + 1$. A piecewise-defined function switches rules depending on where the input lands.

A parking garage is a classic example:

The cost is still a function of hours (every input has exactly one output), but the formula you use depends on the input.

We collect the rules under a single brace:

Key Fact: The brace does not make two functions. It is shorthand for one function with several rules, each paired with the part of the domain where it applies.

Reading the Brace

Every row has two parts: a rule on the left and a condition on the right.

$\underbrace{2x + 1}_{\text{the rule}} \qquad \underbrace{x < 0}_{\text{the condition}}$

To evaluate, ask one question: which condition does my input satisfy? Then use that row's rule and no other.

For $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & x \ge 0 \end{cases}$:

• $x = -3$ satisfies $x < 0$, so use the top rule: $f(-3) = 2(-3) + 1 = -5$.
• $x = 4$ satisfies $x \ge 0$, so use the bottom rule: $f(4) = 4^2 = 16$.
• $x = 0$: is $0 < 0$? No. Is $0 \ge 0$? Yes. Bottom rule: $f(0) = 0$.

The boundary

The symbols $<$ and $>$ exclude the endpoint; $\le$ and $\ge$ include it. At a boundary value, exactly one condition is true: the one with the "or equal to" line.

Conditions must not overlap

If two conditions claimed the same $x$, that input would have two outputs, and $f$ would not be a function. The conditions $x < 0$ and $x \ge 0$ never overlap. (A piecewise function does not have to cover every real number: its domain is simply the union of its conditions.)

Absolute value is piecewise

The absolute value function is the most familiar piecewise-defined function:

$\lvert x \rvert = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$

For example, $g(x) = \lvert x - 3 \rvert$ equals $x - 3$ when $x \ge 3$ and $3 - x$ when $x < 3$: the expression inside is made non-negative by flipping its sign on the interval where it is negative.

Evaluating, Step by Step

Use this function throughout:

Find $f(-4)$

Pick the piece. Is $-4 < -1$? Yes: top rule. Substitute. $f(-4) = -(-4) - 2 = 2$.

Find $f(1)$

Pick the piece. Is $-1 \le 1 < 2$? Yes: middle rule. Substitute. $f(1) = 1^2 = 1$.

Find $f(5)$

Pick the piece. Is $5 \ge 2$? Yes: bottom rule, the constant $6$. So $f(5) = 6$. The input does not appear in a constant rule, so the output is $6$ for every $x \ge 2$.

Find $f(-1)$: a boundary

• Top condition $x < -1$: is $-1 < -1$? No (strict).
• Middle condition $-1 \le x < 2$: is $-1 \le -1$? Yes.

So $f(-1) = (-1)^2 = 1$.

Key Fact: Do not grab the rule whose number "looks close." Read the inequality sign at the boundary.

Which piece applies? 🎯

Evaluate it. ✍️

Use $f(x) = \begin{cases} -x - 2 & x < -1 \\ x^2 & -1 \le x < 2 \\ 6 & x \ge 2 \end{cases}$

1) $f(-4)$:

2) $f(0)$:

3) $f(2)$:

Match each input to its piece. 🔽

For $f(x) = \begin{cases} -x - 2 & x < -1 \\ x^2 & -1 \le x < 2 \\ 6 & x \ge 2 \end{cases}$ and $g(x) = \lvert x - 3 \rvert$:

Key Takeaways — Part 4

• A piecewise-defined function is one function with a rule for each part of its domain
• To evaluate: find the one condition the input satisfies, then use only that rule
• $<$ and $>$ exclude the boundary; $\le$ and $\ge$ include it
• Conditions must not overlap; they need not cover every real number
• $\lvert x \rvert$ is piecewise: $x$ for $x \ge 0$ and $-x$ for $x < 0$

Up Next: Part 5 — graphing with open and closed dots, and building piecewise models from words and data.

🧭 Function Model Selection and Construction

Part 5 of 7 — Building Piecewise Models

Graphing a Piecewise-Defined Function

For each piece:

1. Graph the rule as usual (line, parabola, constant).
2. Keep only the part over that piece's interval.
3. Mark the endpoints: $\le$ or $\ge$ (included) → a closed dot ●; $<$ or $>$ (excluded) → an open dot ○.

Worked Example: the pieces meet

At the boundary $x = 1$, plug $1$ into both rules: the left rule gives $1 + 3 = 4$ (open dot, since $x < 1$ excludes $1$) and the right rule gives $-2(1) + 6 = 4$ (closed dot). Both land at $(1, 4)$, so the closed dot fills the open one and the graph has no gap. The value is $h(1) = 4$.

Worked Example: the graph jumps

At $x = 2$: the left rule gives $2 + 1 = 3$, drawn as an open dot at $(2, 3)$; the right rule gives $2 - 3 = -1$, drawn as a closed dot at $(2, -1)$. The two heights differ, so the graph jumps down by $4$ at $x = 2$. It is still a function: $j(2) = -1$, the closed dot.

Domain and range

For $p(x) = \begin{cases} x + 3 & -3 \le x < 1 \\ 5 & 1 \le x \le 4 \end{cases}$:

• Domain: the union of the conditions, $[-3, 4]$.
• Range: the first piece runs from $p(-3) = 0$ up to, but not including, $1 + 3 = 4$, giving $[0, 4)$; the second piece contributes the single value $5$. Range: $[0, 4) \cup \{5\}$.

Key Fact: A vertical line at any $x$ crosses the graph exactly once: that is the picture of "one output per input."

Read the graph above. 🎯

Building a Piecewise Model from Words

Tiered pricing

A garage charges a flat $5$ dollars for up to $2$ hours and $3$ dollars for each additional hour (charged continuously). Let $C(t)$ be the cost in dollars after $t$ hours.

• For $0 < t \le 2$: the cost is the constant $5$.
• For $t > 2$: start from $5$ and add $3$ per hour beyond $2$: $5 + 3(t - 2)$.

The pieces agree at $t = 2$ (both give $5$), so the cost never jumps.

From data with a change point

A tank's volume $V(t)$, in liters, is recorded every $5$ minutes:

The behavior changes at $t = 10$: the volume rises $100$ liters every $5$ minutes, then falls $40$ liters every $5$ minutes. Model each interval separately.

• Filling, $0 \le t \le 10$: rate $\frac{100}{5} = 20$ L/min, starting at $50$: $V = 50 + 20t$.
• Draining, $10 < t \le 20$: rate $\frac{-40}{5} = -8$ L/min, starting from $250$ at $t = 10$: $V = 250 - 8(t - 10)$.

Both rules give $250$ at $t = 10$, as a tank's volume cannot change instantly. The domain in context is $0 \le t \le 20$, the interval the data cover.

A tax bracket

A simplified tax $T(x)$ on an income of $x$ dollars charges $10\%$ on the first $10{,}000$ dollars and $20\%$ on every dollar above that:

The constant $1000$ is the tax already owed on the first bracket, so the brackets agree at $x = 10000$. For an income of $25{,}000$ dollars: $T(25000) = 1000 + 0.20(15000) = 4000$ dollars.

Key Fact: In a model where the quantity cannot jump (volume, cost, tax), the rules must agree at the change point. Build the second rule from the value where the first one ends.

Build the model. 🎯

Use the tank model. ✍️

$V(t) = \begin{cases} 50 + 20t & 0 \le t \le 10 \\ 250 - 8(t - 10) & 10 < t \le 20 \end{cases}$ (liters, minutes)

1) $V(7)$, in liters:

2) $V(15)$, in liters:

3) The time $t$, in minutes, at which the tank holds the most water:

Tiered pricing. 🔽 A garage charges a flat $5$ dollars for up to $2$ hours and $3$ dollars per additional hour, charged continuously. $C(t)$ is the cost in dollars after $t$ hours.

Key Takeaways — Part 5

• Graph each piece over its own interval; closed dot for an included endpoint, open dot for an excluded one
• Plug the boundary into both rules: equal heights mean the pieces meet, different heights mean a jump
• Domain is the union of the conditions; find the range piece by piece
• From words: a flat fee is a constant piece; "per additional unit beyond $a$" means $\text{(value at } a) + \text{rate} \cdot (t - a)$
• From data: find the change point where the pattern switches, model each interval, and make the rules agree there when the quantity cannot jump

Up Next: Part 6 — the assumptions behind a model, and where it stops being valid.

🧭 Function Model Selection and Construction

Part 6 of 7 — Assumptions, Limitations and Domain Restrictions

Every Model Assumes Something

Choosing a function type is a claim about the situation. Naming that claim is what the exam calls an assumption:

Where a Model Stops Being Valid

• The observed domain. A regression describes the data it was built from. Predicting inside that interval (interpolation) is usually reliable; predicting far outside it (extrapolation) assumes the pattern continues, which may not be true.
• Domain restrictions from context. Lengths, volumes, populations and costs cannot be negative; a model that produces a negative value there is no longer describing the situation.
• Discrete versus continuous inputs. The number of tickets sold is a whole number. A model may treat it as continuous for convenience, but $R(150.5)$ has no meaning in context.
• Units and rounding. A rate from the model must be stated as output units per input unit, and values that come from a rounded regression are approximate.

Key Fact: A limitation names the assumption that could fail: "the model assumes the population keeps growing at a constant $550$ people per year, which may not hold $12$ years past the data."

Worked Examples

A population model

A town's population, in thousands, $t$ years after $2010$:

First differences: $1.1, 1.1, 1.1, 1.1$ thousand per $2$ years, so the rate of change is constant at $0.55$ thousand people per year and a linear model is appropriate:

$P(t) = 12.0 + 0.55t$

Prediction. For $2030$, $t = 20$: $P(20) = 12.0 + 11.0 = 23.0$. The model predicts a population of approximately $23{,}000$ in $2030$.

Limitation. The data end at $t = 8$. The prediction assumes the town keeps growing by exactly $550$ people per year for $12$ more years, which housing, jobs or policy could change.

A candle: the context ends the domain

A candle $25$ cm long burns at a steady $1.5$ cm per hour: $L(t) = 25 - 1.5t$.

The formula gives $L(20) = -5$, but a length cannot be negative. The candle is gone when $L(t) = 0$, at $t = \frac{25}{1.5} \approx 16.667$ hours. The domain in context is $0 \le t \le 16.667$, and the model says nothing after that.

Choosing between two models

A delivery van's distance from the depot, in miles: $0, 30, 60, 90, 120$ at $t = 0, 1, 2, 3, 4$ hours.

• A student proposes a quadratic because "distance is growing." But the first differences are $30, 30, 30, 30$: constant.
• A linear model, $d(t) = 30t$, matches the pattern: the van travels at a steady $30$ miles per hour.

The quadratic is not wrong because it is more complicated; it is wrong because its signature (changing rate of change) is not in the data.

When a piecewise model is needed

If the next readings were $120, 120, 120$ at $t = 5, 6, 7$, one line could no longer describe the trip: the van stopped. The pattern changed at $t = 4$, so a piecewise model, $30t$ for $0 \le t \le 4$ and $120$ for $t > 4$, is appropriate.

Write the justification. 🔽 Choose the sentence that correctly justifies or limits each model.

Assumptions and restrictions. 🎯

Domain from context. ✍️

A candle $25$ cm long burns steadily: $L(t) = 25 - 1.5t$ (centimeters, hours). A town's population is $P(t) = 12.0 + 0.55t$ (thousands, years after $2010$).

1) $L(4)$, in centimeters:

2) The time, in hours, at which the candle is completely burned (the right endpoint of the domain in context):

3) The population, in thousands, the model predicts for $2030$:

Key Takeaways — Part 6

• Every model type assumes a pattern: constant rate (linear), constant second differences (quadratic), a switch at a change point (piecewise)
• A model is trusted on the observed domain; extrapolation assumes the pattern continues, and that is the limitation to state
• The domain in context ends where the output stops making sense (negative length, negative volume)
• Some inputs are discrete; the model's non-integer inputs may have no meaning
• Choose between two models by which signature the data show, not by which formula is fancier
• Values from a model are approximate, with units

Up Next: Part 7 — a full FRQ-style problem, then the mastery check.

🧭 Function Model Selection and Construction

Part 7 of 7 — FRQ Workshop and Mastery Check

Anatomy of FRQ Question 2: Modeling a Non-Periodic Context

This free-response question is calculator-active. It gives a table (or a description) of a real quantity and asks you to build and use a model. The parts almost always follow this script:

Key Fact: Every sentence you write should be checkable: a number, its units, the interval it applies to, and the reason it is true.

Worked FRQ

The table gives the number of visitors, in thousands, to a new museum during month $t$ after it opened.

(A) Choose and justify a model type; state the model

First differences: $1.4, 2.3, 3.2, 3.9, 4.8$. They are not constant, so the model is not linear.

Second differences: $0.9, 0.9, 0.7, 0.9$: roughly constant.

Justification: "The first differences increase by roughly the same amount each month (the second differences are roughly constant, about $0.9$), so the rate of change is changing at a roughly constant rate and a quadratic model is appropriate."

Quadratic regression on the calculator:

$N(t) = 0.418t^2 + 0.201t + 7.560$

(B) Predict, with units and "approximately"

Using the stored full-precision model, $N(8) \approx 35.909$.

Answer: "The model predicts that approximately $35.909$ thousand visitors, about $35{,}900$ people, will visit the museum in month $8$."

(Using the rounded coefficients gives $0.418(64) + 0.201(8) + 7.560 = 35.920$: a different third decimal. Keep full precision.)

(B) Interpret an average rate of change

From the table, on $[2, 6]$:

$\frac{N(6) - N(2)}{6 - 2} = \frac{23.8 - 9.6}{4} = \frac{14.2}{4} = 3.55$

Answer: "Between month $2$ and month $6$, the number of monthly visitors increased at an average rate of $3.55$ thousand visitors per month."

(C) A limitation

$N(12) \approx 70.140$. Answer: "The model assumes the pattern of increasing growth continues past month $6$. A museum's monthly attendance cannot grow faster and faster indefinitely (building capacity, the novelty of a new museum fading), so predictions far beyond the data, such as month $12$, are unreliable."

Key Fact: Notice what each answer contains: a number, its units, the interval or input it belongs to, and for the model type, the pattern that justifies it.

FRQ checks. 🎯 Use the museum data: $N = 8.2, 9.6, 11.9, 15.1, 19.0, 23.8$ thousand visitors in months $1$ through $6$.

Write the answer sentence. 🔽 The museum model gives $N(8) \approx 35.909$ and $N(12) \approx 70.140$ (thousand visitors).

Rates from the table. ✍️

1) The average rate of change of $N$ over $[1, 6]$, in thousand visitors per month:

2) The average rate of change of $N$ over $[5, 6]$, in thousand visitors per month:

3) The second difference between the first two first differences ($1.4$ and $2.3$):

Function Model Selection and Construction — Summary 🎓

The FRQ Checklist

1. Type + pattern: "roughly constant first/second differences," "a product of lengths," "the rule changes at …"
2. Model: regression equation to three decimal places; full precision afterward
3. Prediction: "approximately," units, the input it belongs to
4. Rate of change: value, output units per input unit, interval
5. Assumption or limitation: the pattern assumed and where it may fail (extrapolation, negative quantities, discrete inputs)

Key Fact: A model is a claim about a pattern. Choose it by the pattern, use it with units, and say where it stops being true.

Exit Quiz ✅