Knowledge Notes | Carey AP Precalculus Practice Lab

Use the model only on its stated domain, with quantities and units attached.

What a strong response shows

Define the input, output and units.
Use the given relationships to determine parameters.
Check the physical domain and verify a known value.

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What You Need to Know

Key idea

A function model is a mathematical claim about how an input controls an output. A strong AP response names the variables, selects a function family that matches the evidence, states the contextual domain, and explains what the result means in the situation.

Definitions

The input (independent variable) is the quantity you choose or observe first. The output (dependent variable) is the quantity predicted by the model. A parameter is a number in the formula whose value is determined by a restriction, a data pair, or a regression calculation.

Method

Step $1$ : identify the input, output, units, and allowed values. Step $2$ : look for constant differences, a constant product, a threshold, or a trend in a table or graph. Step $3$ : write the model and determine its parameters. Step $4$ : evaluate, check the domain, and interpret with units.

\text{average rate of change}=\frac{f(b)-f(a)}{b-a}

Worked example

If a quantity rises by the same amount whenever the input increases by one unit, begin with a linear model $y=m x+b$ . Use one known point to find $b$ after finding the constant rate $m$ . Then check another data point before making a prediction.

Common mistake

A formula can fit a few numbers and still be unreasonable outside the context. Do not extrapolate beyond the observed interval without explaining the assumption, and do not omit units from a rate or a final value.

Mini check

Before calculating, can you answer all four questions: What is $x$ ? What is $y$ ? What units do they use? Which x-values are allowed?

Learning Objectives

1.14.A

Construct a linear, polynomial, regression, or piecewise model from restrictions, a table, a graph, or a verbal rule. Success means your equation agrees with the evidence and your domain matches the context.

1.14.B

Construct and apply inverse and inverse-square rational models. Success means you preserve the reciprocal relationship and keep the square on the distance when the situation is inverse-square.

1.14.C

Use a model to evaluate, predict, compare rates, and interpret an answer with units. Success means you distinguish a value, a rate, and elapsed time instead of reporting the wrong quantity.

Practice path

Questions 1–2 emphasize model construction and interpretation; Questions 3–4 use inverse relationships; Questions 5–6 test recoding and threshold rules; Question $7$ tests elapsed time.

Mini check

Point to one practice question for each objective. If you cannot explain why it belongs there, return to the Key Concepts card before attempting it.

Key Concepts

Linear model

A linear model $y=m x+b$ has a constant rate of change $m$ . The graph is a line, and the sign of $m$ tells whether the output increases or decreases as the input increases.

Piecewise model

A piecewise model uses different formulas on different intervals. A threshold often creates a flat first piece and a second piece that charges only for the amount beyond the threshold.

Inverse model

In $y=\frac{k}{x}$ , the product $x y$ is constant. Doubling $x$ halves $y$ , provided the context keeps the same constant $k$ .

Inverse-square model

In $y=\frac{k}{x^{2}}$ , the product $x^{2} y$ is constant. Doubling $x$ makes $y$ one-fourth as large; this is different from ordinary inverse proportionality.

Regression model

A regression equation summarizes a trend in data. Its coefficients depend on how the input is coded, so translate a coded input back to the real year, distance, or time before interpreting it.

y=mx+b,\qquad y=\frac{k}{x},\qquad y=\frac{k}{x^2},\qquad C(d)=\begin{cases}C_0,&0\le d\le d_0\\C_0+r(d-d_0),&d>d_0\end{cases}

Common mistake

The phrases “inversely proportional” and “decreases as $x$ increases” are not interchangeable. Use the formula or the constant-product test to identify the relationship.

Mini check

If $x$ is multiplied by $3$ , what happens to $y$ in an inverse model? What happens in an inverse-square model? Answer: $y$ is divided by $3$ versus divided by $9$ .

Relevant Representations

Representation map

A table is useful for detecting repeated differences, products, or ratios. An equation makes restrictions and parameters visible. A graph reveals shape, intercepts, thresholds, and whether a prediction is interpolation or extrapolation.

Use the representation that answers the question
Representation	Look for	AP interpretation
Table	constant differences or products	identify a function family
Equation	parameters, domain, units	evaluate or solve
Graph	shape, threshold, intercept	describe behavior
Words	included amount, rate, restriction	write the correct piece or domain

Recoded input

If $t=\text{year}-2000$ , then $t=24$ represents the year $2024$ . The model is not wrong when the intercept changes after recoding; the meaning of the input has changed.

t=\text{year}-2000\quad\Longrightarrow\quad\text{calendar year}=2000+t

Recoded input changes the intercept, not the datadata trendmodel

Worked example

A data table indexed by $t=0,\,1,\,2,\,\dots$ is not indexed by calendar year. If a regression predicts $y=74$ at $t=6$ , report the prediction for the year $2006$ , not the year $6$ .

Common mistake

Do not read a graph scale as if every grid line were one unit. Read the labels, then check that the plotted point lies in the contextual domain.

Mini check

Which representation would you choose to verify that a pricing rule is continuous at a threshold? Write the two pieces and compare their values at the endpoint.

Essential Relationships or Formulas

Constant rate

For a linear relationship, $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ . Keep the order consistent in numerator and denominator so the sign is meaningful.

Inverse constant

For $y=\frac{k}{x}$ , calculate $k=x y$ from any reliable data pair. For $y=\frac{k}{x^{2}}$ , calculate $k=x^{2} y$ ; the exponent belongs to $x$ before you solve for the constant.

Threshold rule

If $d_{0}$ units are included, the excess is $d-d_{0}$ . The second piece charges $r\left(d-d_{0}\right)$ , not rd. This preserves the included amount and makes the rule continuous when appropriate.

Elapsed time

Solve for the two input values first. Then subtract the earlier input from the later input; do not report the later input alone.

m=\frac{y_2-y_1}{x_2-x_1},\qquad k=xy\text{ or }k=x^2y,\qquad \Delta t=t_2-t_1

C(d)=C_0+r(d-d_0)\quad(d>d_0)

Worked example

Suppose $I=26.5$ when $x=4$ in $I=\frac{k}{x^{2}}$ . Then $k=26.5 \left(4^{2}\right)=424$ . At $x=3.3$ , $I=\frac{424}{3.3^{2}}\approx 38.9$ . The answer is larger because the distance is smaller.

Common mistake

Do not round $k$ before the final evaluation when the question asks for a decimal approximation. Keep the exact constant or full calculator precision until the end.

Mini check

For a model $y=\frac{120}{x^{2}}$ , compare $y$ at $x=2$ and $x=6$ . The second value should be $\frac{1}{9}$ of the first because $6^{2}$ is nine times $2^{2}$ .

Worked Examples

Example 1 · inverse-square

A light source has intensity $26.5$ at distance $4$ . Assume $I=\frac{k}{x^{2}}$ . First compute $k=26.5\cdot 4^{2}=424$ . Next evaluate $I \left(3.3\right)=\frac{424}{3.3^{2}}\approx 38.9$ . Finally state the interpretation: moving closer increases the modeled intensity.

Inverse-square model: intensity increases as distance decreases

Example 2 · threshold pricing

A plan includes $5$ GB for $79.95$ dollars and charges $10$ dollars per additional gigabyte. For $0\le d\le 5$ , $C\left(d\right)=79.95$ . For $d>5$ , $C\left(d\right)=79.95+10 \left(d-5\right)$ . At $d=8$ , $C\left(8\right)=109.95$ dollars.

Threshold pricing is a piecewise rule

Example 3 · coded regression

A model uses $t=\text{year}-2010$ . If the calculator output at $t=6$ is $74$ , the prediction belongs to $2016$ . The coefficient describes change per coded year, while the intercept describes the model at year $2010$ .

Common mistake

A correct numerical substitution can still answer the wrong question. Label every intermediate value with its meaning before writing the final sentence.

Mini check

Which step would you show first in each example: identify the model family, find the constant, evaluate, or interpret? The family and variables must come before calculator entry.

Common Mistakes

Mistake 1 · wrong relationship

Using $y=k x$ when the evidence is inverse, or using $y=\frac{k}{x}$ when the evidence is inverse-square, changes every prediction. Test $x y$ or $x^{2} y$ with two data pairs before choosing.

Mistake 2 · lost exponent

Writing $k=x y$ for an inverse-square situation omits the square. Circle the word “square” in the prompt and keep it visible in the model.

Mistake 3 · double charging

In a threshold plan, the included amount is already paid for. Use the excess $d-d_{0}$ in the second piece; never use $d$ unless the problem explicitly says the entire amount is charged at the higher rate.

Mistake 4 · coded input

A regression with $t=\text{year}-2000$ does not use the calendar year directly. Convert the requested year to $t$ before evaluating, and convert $t$ back when interpreting.

Mistake 5 · wrong time quantity

A target time and an elapsed time are different. Solve for both target inputs and subtract them if the question asks “how long” or “how many hours.”

Diagnostic checklist

Before submitting, underline the model family, circle the domain, box the units, and write one sentence explaining the output. If one of these is missing, the work is not finished.

Mini check

A student writes $C\left(d\right)=79.95+10 d$ for a plan that includes $5$ GB. Identify the error and repair the formula for $d>5$ : $C\left(d\right)=79.95+10 \left(d-5\right)$ .

AP Exam Tips

Set up before calculating

Write the variable definitions and the model before entering numbers. A calculator decimal without a model does not show what was evaluated or whether the input was coded correctly.

Use units as a guardrail

Rates carry units such as dollars per gigabyte or intensity per distance. Keep those units beside the expression; they reveal when you divide in the wrong direction.

Interpret the requested quantity

“What is the value?” asks for one evaluation. “How fast?” asks for a rate. “How long?” asks for a difference between two inputs. Match the final sentence to the verb in the prompt.

Check endpoints and domain

For piecewise rules, test the threshold from both sides. For rational models, reject inputs that are impossible in context or that make a denominator zero.

Calculator communication

When a graphing calculator is allowed, record the model, the input, the unrounded display value, and the rounded answer with units. This makes a numerical response auditable.

Mini check

If a calculator gives $t_{1}=2.4$ and $t_{2}=7.9$ for two target outputs, report $5.5$ time units when the prompt asks for elapsed time, not $7.9$ .

Final review

Ask: Is the function family justified? Is the domain stated? Are the units and coded inputs correct? Does the answer describe the situation rather than only display a number?

TRY IT IN A NEW SETTING

Can you transfer the method?

An inverse-square model has $I(r)=k/r^2$ and $I(2)=12$ . Find $k$ and predict $I \left(4\right)$ . State a meaningful distance domain.

Compare your reasoning

From $12=\frac{k}{4}$ , $k=48$ . Thus $I \left(4\right)=\frac{48}{16}=3$ . A meaningful distance domain is $r>0$ , subject to the context and the range over which the model is reasonable.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.14 · Function Model Construction and Application

Topic 1.14 · Function Model Construction and Application

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?