Knowledge Notes | Carey AP Precalculus Practice Lab

Patterns support a model under its assumptions; genuine exponential growth or decay uses a positive ratio other than 1. — Patterns support a model under its assumptions; genuine exponential growth or decay uses a positive ratio other than $1$ .

What a strong response shows

State the input condition together with the output pattern.
Compare the candidate family with the context.
Separate finite-data support from proof or guaranteed extrapolation.

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What You Need to Know

Key idea

A model is a rule that summarizes a pattern. Start with evidence, then choose the simplest family that matches both the data and the story; the family alone is never the whole answer.

Fast decision rule

Equal input steps are required before using differences. A constant first difference supports linear; a first nonzero constant second difference supports quadratic; a first nonzero constant third difference supports cubic. A near-constant ratio supports exponential behavior.

Worked steps

$1$ . Name the input and output, with units. $2$ . Check whether inputs are equally spaced. $3$ . Test differences, ratios, shape, or a rule change. $4$ . State the allowed inputs and the assumption behind extending the pattern.

Use the table of values of [function] to determine if [function] is best modeled by a linear, quadratic, exponential, or logarithmic function.

Common mistake

A box problem needs positive remaining lengths, not just a formula. For the 11.2-inch side, $11.2-2 x>0$ , so the cut size has an upper limit. The picture and the context decide the domain.

Mini check

Before choosing a family, can you point to the evidence, say whether the inputs are equally spaced, and write the interval where the rule makes sense?

\Delta y=y_{k+1}-y_k\qquad\Delta^2 y=\Delta(\Delta y)\qquad\Delta^3 y=\Delta(\Delta^2 y)

Learning Objectives

Key idea

$1.13$ asks you to identify a family from a scenario, table, graph, or change description, then explain why the choice is reasonable.

Fast decision rule

Separate “what is changing?” from “what is being measured?” A postal graph may use weight as input and price as output; a daily-cookie graph may show the daily increase, not the total number of cookies.

Worked steps

$1$ . Label input and output. $2$ . Find the visible pattern. $3$ . Choose the smallest family that explains it. $4$ . State a domain, units, and assumption.

Give a reason for your answer in part $C$ $\left(i\right)$ based on the relationship between the change in the output values of [function] and the change in the input values of [function]. Refer to the values in the table in your reasoning.

Common mistake

A correct family with no evidence is incomplete. Also do not confuse a rate, such as cookies added per day, with the total being modeled.

Mini check

For one question, underline the input and output first. What observation would make you reject the tempting family?

Key Concepts

Key idea

Match the test to the data: differences compare equal additions, ratios compare equal multipliers, and a piecewise model handles a visible change of rule.

Fast decision rule

With equal spacing, constant first differences mean linear. Constant second differences mean quadratic only when that second difference is the first nonzero constant level; constant zero second differences are still linear. Constant third differences support cubic when lower levels are not constant.

Worked example

The tetrahedral numbers $1,\,4,\,10,\,20,\,35,\,56,\,84$ have equal input steps and third differences $1,\,1,\,1,\,1$ , so a cubic pattern is supported. That conclusion comes from the first nonzero constant difference, not from the fact that the numbers increase.

Give a reason for your answer in part $C$ $\left(i\right)$ based on the definition of a function and the table of values of [function]. Refer to the values in your reasoning.

Common mistake

Do not call every curved or increasing pattern quadratic. A constant first difference has second difference $0$ , so it is linear; a quadratic needs a nonzero constant second difference.

Mini check

Which test fits each case: postal price jumps at ounce boundaries, crowd capacity from square-field area, and $1,\,4,\,10,\,20,\,35,\,56,\,84$ ?

Evidence $\to$ model family
Evidence	Supported family
constant first difference	linear
constant second difference	quadratic
constant third difference	cubic
constant ratio	exponential
two stable rates	piecewise linear

Relevant Representations

Key idea

Words provide the story and restrictions, tables expose repeated changes, graphs show shape and breakpoints, and equations make the rule explicit. Use more than one representation when a distractor can hide the meaning.

Fast decision rule

Read axis labels before reading the curve. In the cookie question, the plotted value is the increase from the previous day; it is not the total cookie count. A roughly linear increase in daily additions leads to a quadratic total.

Worked steps

$1$ . Read both axes and units. $2$ . Check whether endpoints are included. $3$ . Decide whether the output is a level, rate, or change. $4$ . Use the context to limit the domain.

Use the table of values of [function] to determine if [function] is best modeled by a linear, quadratic, exponential, or logarithmic function.

Common mistake

Grid spacing is not automatically one unit, and an open endpoint does not mean the value is allowed. Read labels and endpoint symbols.

Mini check

If a piecewise-linear model changes slope at both $x=2$ and $x=4$ , how many line pieces are needed across an interval containing both breakpoints? Answer: three pieces.

A slope change at $x=2$ and another at $x=4$ creates three line pieces across an interval that includes both breakpoints.

Model evidence from equal-step changes

Essential Relationships or Formulas

Key idea

Formulas encode the reason a family fits. A geometric restriction can be stricter than the algebraic formula, and a discrete input can restrict the output values.

Fast decision rule

For an open box, write each remaining side as original length minus $2 x$ and require every length to be positive. For a square field, doubling side length multiplies area, and roughly constant people-per-area multiplies capacity by four.

Worked steps

$1$ . Define variables and units. $2$ . Translate each physical length or rate. $3$ . Build the model. $4$ . Check endpoints, positivity, and whether inputs are continuous or discrete.

The model is an appropriate model for [context]. Based on this information, what is the largest value of [input] for which the model is appropriate? Give a reason for your answer.

Common mistake

Do not use the total amount when the prompt gives a rate or increment, and do not ignore a physical upper bound just because the formula accepts more numbers.

Mini check

What restriction comes from $11.2-2 x>0$ ? What changes when the input is weight but the output is a stepwise price?

\Delta y=y_{k+1}-y_k\qquad\Delta^2y=\Delta(\Delta y)\qquad\Delta^3y=\Delta(\Delta^2y)

y=mx+b\qquad y=ax^2+bx+c\qquad y=ax^3+bx^2+cx+d

Worked Examples

Key idea

Build the relationship before calculating. Each of the six Questionbank items tests a different modeling clue.

Fast decision rule

$Q 1$ : positive box lengths set the maximum cut. $Q 2$ : ounce steps create discrete prices. $Q 3$ : square area makes capacity quadratic. $Q 4$ : two slope changes call for two line segments. $Q 5$ : a nearly linear daily increase makes the total quadratic. $Q 6$ : equal-step tetrahedral data with constant third differences support cubic.

Worked example

For $Q 1$ , $V=x \left(11.2-2 x\right) \left(13.3-2 x\right)$ , and $0<x<5.6$ . For $Q 3$ , side length $s$ gives area $s^{2}$ , so capacity is proportional to $s^{2}$ when people per square meter is similar. For $Q 6$ , the third differences of $1,\,4,\,10,\,20,\,35,\,56,\,84$ are $1,\,1,\,1,\,1$ .

Explain how this information can be used to determine the domain limitations for the model.

Common mistake

Do not treat a graph of increments as a graph of totals, and do not call a sequence cubic until the equal spacing and first nonzero constant third difference are checked.

Mini check

Pick one of Q1–Q6 and write the evidence, the selected family, and the restriction in one sentence.

V(x)=4x^3-49x^2+148.96x\qquad 0<x<5.6

\Delta y:5,9,13,17\qquad\Delta^2y:4,4,4

Common Mistakes

Key idea

Most wrong choices come from misreading the quantity or applying a test outside its conditions.

Fast decision rule

Ask: Is the input equally spaced? Is the output a total or a change? Is the rule continuous or stepwise? Does the story impose a positive length, a discrete price, or a finite interval?

Worked steps

$1$ . Re-label axes or variables. $2$ . Write one concrete pair of values. $3$ . Test the smallest useful pattern. $4$ . Reject choices that violate units or the situation.

Common mistake

“It increases” is not evidence for exponential or quadratic. A zero second difference belongs to a linear pattern, and a third-difference claim needs equal input spacing.

Mini check

If a student says “the tetrahedral numbers are increasing, so they are exponential,” what missing test would you ask for?

AP Exam Tips

Key idea

Write a short evidence sentence before choosing an equation. AP readers can follow your model when input, output, units, and domain are named.

Fast decision rule

Evidence $\to$ quantity $\to$ family $\to$ restriction $\to$ interpretation. For differences, add “inputs are equally spaced” and identify the first nonzero constant level.

Worked steps

$1$ . Quote the useful feature (positive lengths, step prices, square area, slope changes, daily additions, or third differences). $2$ . Explain why it maps to the family. $3$ . State the domain and units. $4$ . Check whether the prediction is interpolation or extrapolation.

The model is an appropriate model for [context]. Based on this information, what is the largest value of [input] for which the model is appropriate? Give a reason for your answer.

Common mistake

A calculator result cannot repair a wrong variable, wrong family, or wrong domain. Keep the exact relationship visible before rounding.

Mini check

Can you explain Q1–Q6 in six different evidence sentences without saying only “because it is increasing”?

TRY IT IN A NEW SETTING

Can you transfer the method?

At inputs $0$ , $1$ , $2$ , $3$ , the outputs are $2$ , $6$ , $18$ , $54$ . Choose a model family, give one matching rule and state a limitation.

Compare your reasoning

Equal input steps of $1$ give constant output ratios of $3$ , so an exponential model is supported. One matching rule is $y=2\cdot 3^{x}$ . Assuming this pattern continues allows prediction; the finite table does not prove how an arbitrary function behaves between or beyond the observations.

Use it in a real question

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

Practise this topic Check the whole unit

1.13 · Function Model Selection and Assumption Articulation

Topic 1.13 · Function Model Selection and Assumption Articulation

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?