AP Precalculus Unit 1: Polynomial and Rational Functions Notes

Introduction

Use covariation and rates to distinguish polynomial structures.

Preserve rational restrictions while identifying zeros, holes, and asymptotes.

Select and construct models from structural evidence and explicit assumptions.

Describe

Name how outputs covary with inputs.

Represent

Choose an equation, table, or graph that exposes structure.

Defend

State restrictions and model assumptions.

Reference

Formula and theorem sheet

Match each expression with the condition that makes it valid.

UseExpressionCondition

Average rate

[f(b)-f(a)]/(b-a)

a\ne b

Polynomial ends

f(x)\sim a_nx^n

Use degree parity and leading sign.

Rational zeros

P=0, Q\ne0

Preserve original restrictions.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 4 free-response questions. Units 1–3 are assessed. Unit 4 is taught here as course extension material and is not assessed on the AP Exam.

1.1

Change in Tandem

What decides a correct solution for Change in Tandem?

Short answer

Covariation describes how two quantities change together, including direction and relative pace.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Change in Tandem

Track paired inputs and outputs rather than treating either column alone. Equal input steps make changes, ratios, and reversals easier to compare.

A relationship can change direction or rate across its domain. Context determines whether interpolation or extrapolation is defensible. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\Delta y=f(x+\Delta x)-f(x)

The output change is tied to a specified input change.

Use when: Both values use the same function and compatible units.
Do not assume: A positive output does not imply positive covariation; direction comes from the change.

Change in Tandem: the useful representation

Covariation describes how two quantities change together, including direction and relative pace. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Change in Tandem: the useful representation: values

Case	Result
Key idea	Covariation describes how two quantities change together, including direction and relative pace.
Boundary	Both values use the same function and compatible units.

Use the structure in Change in Tandem

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Covariation describes how two quantities change together, including direction and relative pace.
Both values use the same function and compatible units.
Failure check: A positive output does not imply positive covariation; direction comes from the change.

1.2

Rates of Change

What decides a correct solution for Rates of Change?

Short answer

Average rate of change is a secant slope with output units per input unit.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rates of Change

Subtract outputs in the same order as inputs, then divide by the input change. The sign communicates direction and the magnitude communicates pace.

An average rate summarizes an interval; it need not equal the rate at every point. A zero denominator makes the quotient invalid. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\frac{f(b)-f(a)}{b-a}

The quotient measures output change per input unit.

Use when: $a\ne b$ and both inputs lie in the domain.
Do not assume: Dividing by $b$ instead of $b-a$ loses the interval scale.

Rates of Change: the useful representation

Average rate of change is a secant slope with output units per input unit. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rates of Change: the useful representation: values

Case	Result
Key idea	Average rate of change is a secant slope with output units per input unit.
Boundary	$a\ne b$ and both inputs lie in the domain.

Use the structure in Rates of Change

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Average rate of change is a secant slope with output units per input unit.
$a\ne b$ and both inputs lie in the domain.
Failure check: Dividing by $b$ instead of $b-a$ loses the interval scale.

1.3

Rates of Change in Linear and Quadratic Functions

What decides a correct solution for Rates of Change in Linear and Quadratic Functions?

Short answer

Linear functions have constant average rate; quadratic rates change linearly with location.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rates of Change in Linear and Quadratic Functions

For a line, every secant slope equals its coefficient of $x$ . For a quadratic, equal input intervals produce changing first differences and constant second differences.

Constant second difference depends on equally spaced inputs. A quadratic can have zero average rate across a symmetric interval without being constant. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\frac{f(x+h)-f(x)}h=2ax+ah+b

A quadratic’s average rate is linear in the starting input.

Use when: $h\ne0$ and $f(x)=ax^2+bx+c$ .
Do not assume: Constant first differences identify a linear model, not a quadratic one.

Rates of Change in Linear and Quadratic Functions: the useful representation

Linear functions have constant average rate; quadratic rates change linearly with location. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rates of Change in Linear and Quadratic Functions: the useful representation: values

Case	Result
Key idea	Linear functions have constant average rate; quadratic rates change linearly with location.
Boundary	$h\ne0$ and $f(x)=ax^2+bx+c$ .

Use the structure in Rates of Change in Linear and Quadratic Functions

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Linear functions have constant average rate; quadratic rates change linearly with location.
$h\ne0$ and $f(x)=ax^2+bx+c$ .
Failure check: Constant first differences identify a linear model, not a quadratic one.

1.4

Polynomial Functions and Rates of Change

What decides a correct solution for Polynomial Functions and Rates of Change?

Short answer

Polynomial degree controls how many rounds of finite differences are needed to become constant.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Polynomial Functions and Rates of Change

Equal-step tables reveal structure: a degree- $n$ polynomial has constant $n$ th finite differences. On a graph, turning and end behavior provide complementary evidence.

Finite-difference evidence assumes equal spacing and can be distorted by rounding. It suggests degree but must agree with the function’s algebraic structure. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\Delta^n f=\text{constant}

The $n$ th equal-step difference detects degree $n$ .

Use when: Inputs are equally spaced and the leading coefficient is nonzero.
Do not assume: Counting visible turns alone may underestimate degree because some turns need not occur.

Polynomial Functions and Rates of Change: the useful representation

Polynomial degree controls how many rounds of finite differences are needed to become constant. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Polynomial Functions and Rates of Change: the useful representation: values

Case	Result
Key idea	Polynomial degree controls how many rounds of finite differences are needed to become constant.
Boundary	Inputs are equally spaced and the leading coefficient is nonzero.

Use the structure in Polynomial Functions and Rates of Change

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Polynomial degree controls how many rounds of finite differences are needed to become constant.
Inputs are equally spaced and the leading coefficient is nonzero.
Failure check: Counting visible turns alone may underestimate degree because some turns need not occur.

1.5

Polynomial Functions and Complex Zeros

What decides a correct solution for Polynomial Functions and Complex Zeros?

Short answer

Nonreal zeros of a real-coefficient polynomial occur in conjugate pairs.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Polynomial Functions and Complex Zeros

Use the factor theorem to move between zeros and factors. If $a+bi$ is a zero and coefficients are real, then $a-bi$ supplies the paired quadratic factor.

Multiplicity affects whether a real graph crosses or touches the axis. Complex zeros do not appear as real $x$ -intercepts. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

a+bi\Rightarrow a-bi

Conjugate pairing keeps polynomial coefficients real.

Use when: The polynomial has real coefficients.
Do not assume: Drawing a complex zero as a real intercept confuses two number systems.

Polynomial Functions and Complex Zeros: the useful representation

Nonreal zeros of a real-coefficient polynomial occur in conjugate pairs. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Polynomial Functions and Complex Zeros: the useful representation: values

Case	Result
Key idea	Nonreal zeros of a real-coefficient polynomial occur in conjugate pairs.
Boundary	The polynomial has real coefficients.

Use the structure in Polynomial Functions and Complex Zeros

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Nonreal zeros of a real-coefficient polynomial occur in conjugate pairs.
The polynomial has real coefficients.
Failure check: Drawing a complex zero as a real intercept confuses two number systems.

1.6

Polynomial Functions and End Behavior

What decides a correct solution for Polynomial Functions and End Behavior?

Short answer

Degree parity and leading-coefficient sign determine polynomial end behavior.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Polynomial Functions and End Behavior

At large magnitude, the leading term dominates all lower-degree terms. Even degree sends both ends the same way; odd degree sends them in opposite directions.

End behavior does not locate finite turning points or intercepts. A graphing window that is too narrow can hide the true dominant pattern. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

f(x)\sim a_nx^n

The leading term controls both tails.

Use when: $a_n\ne0$ and $|x|$ is large.
Do not assume: Using the constant term to predict the tails ignores dominance.

Polynomial Functions and End Behavior: the useful representation

Degree parity and leading-coefficient sign determine polynomial end behavior. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Polynomial Functions and End Behavior: the useful representation: values

Case	Result
Key idea	Degree parity and leading-coefficient sign determine polynomial end behavior.
Boundary	$a_n\ne0$ and $\|x\|$ is large.

Use the structure in Polynomial Functions and End Behavior

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Degree parity and leading-coefficient sign determine polynomial end behavior.
$a_n\ne0$ and $|x|$ is large.
Failure check: Using the constant term to predict the tails ignores dominance.

1.7

Rational Functions and End Behavior

What decides a correct solution for Rational Functions and End Behavior?

Short answer

Relative numerator and denominator degrees determine horizontal or polynomial end behavior.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rational Functions and End Behavior

Divide by the dominant denominator power or perform polynomial division. A proper remainder vanishes at the ends, leaving the quotient as the asymptotic model.

End behavior is separate from vertical restrictions. A slant or polynomial asymptote may be crossed at finite inputs. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)}

Polynomial division isolates the end-behavior quotient.

Use when: $\deg R<\deg Q$ after division.
Do not assume: Calling every rational tail horizontal ignores numerator degree.

Rational Functions and End Behavior: the useful representation

Relative numerator and denominator degrees determine horizontal or polynomial end behavior. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rational Functions and End Behavior: the useful representation: values

Case	Result
Key idea	Relative numerator and denominator degrees determine horizontal or polynomial end behavior.
Boundary	$\deg R<\deg Q$ after division.

Use the structure in Rational Functions and End Behavior

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Relative numerator and denominator degrees determine horizontal or polynomial end behavior.
$\deg R<\deg Q$ after division.
Failure check: Calling every rational tail horizontal ignores numerator degree.

1.8

Rational Functions and Zeros

What decides a correct solution for Rational Functions and Zeros?

Short answer

A rational function is zero where its simplified numerator is zero and its denominator remains nonzero.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rational Functions and Zeros

Factor numerator and denominator, record restrictions, and cancel common factors only after preserving their excluded inputs. Remaining numerator zeros produce intercepts.

A canceled zero becomes a hole, not an intercept. Multiplicity still controls crossing or touching for zeros that remain. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\frac{P(x)}{Q(x)}=0\iff P(x)=0,\ Q(x)\ne0

A fraction vanishes through its numerator.

Use when: The input remains in the original domain.
Do not assume: Solving only the numerator can include an excluded common-factor zero.

Rational Functions and Zeros: the useful representation

A rational function is zero where its simplified numerator is zero and its denominator remains nonzero. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rational Functions and Zeros: the useful representation: values

Case	Result
Key idea	A rational function is zero where its simplified numerator is zero and its denominator remains nonzero.
Boundary	The input remains in the original domain.

Use the structure in Rational Functions and Zeros

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

A rational function is zero where its simplified numerator is zero and its denominator remains nonzero.
The input remains in the original domain.
Failure check: Solving only the numerator can include an excluded common-factor zero.

1.9

Rational Functions and Vertical Asymptotes

What decides a correct solution for Rational Functions and Vertical Asymptotes?

Short answer

Uncanceled denominator zeros are vertical-asymptote candidates whose one-sided signs describe the branches.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rational Functions and Vertical Asymptotes

Factor first to separate holes from persistent denominator zeros. Then make a sign chart on each side to determine whether outputs approach positive or negative infinity.

A denominator zero alone does not prove an asymptote because cancellation may remove the unbounded behavior. One-sided signs need not match. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

Q(a)=0,\ P(a)\ne0\Rightarrow x=a\text{ is a vertical asymptote}

A surviving denominator zero drives unbounded magnitude.

Use when: The factors are simplified and the numerator stays nonzero at $a$ .
Do not assume: Labeling a canceled factor as an asymptote misclassifies a hole.

Rational Functions and Vertical Asymptotes: the useful representation

Uncanceled denominator zeros are vertical-asymptote candidates whose one-sided signs describe the branches. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rational Functions and Vertical Asymptotes: the useful representation: values

Case	Result
Key idea	Uncanceled denominator zeros are vertical-asymptote candidates whose one-sided signs describe the branches.
Boundary	The factors are simplified and the numerator stays nonzero at $a$ .

Use the structure in Rational Functions and Vertical Asymptotes

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Uncanceled denominator zeros are vertical-asymptote candidates whose one-sided signs describe the branches.
The factors are simplified and the numerator stays nonzero at $a$ .
Failure check: Labeling a canceled factor as an asymptote misclassifies a hole.

1.10

Rational Functions and Holes

What decides a correct solution for Rational Functions and Holes?

Short answer

A common factor creates a removable hole at the excluded input and simplified output height.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Rational Functions and Holes

Factor and cancel to find the nearby rule, but keep the original restriction. Evaluate the simplified expression at the excluded input to locate the open point.

The limit at a hole can exist even though the original function value does not. Filling the point changes the function definition, not its nearby rule. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\frac{(x-a)P(x)}{(x-a)Q(x)}=\frac{P(x)}{Q(x)},\quad x\ne a

Cancellation identifies behavior around the excluded input.

Use when: $Q(a)\ne0$ after cancellation.
Do not assume: Erasing $x\ne a$ falsely fills the hole.

Rational Functions and Holes: the useful representation

A common factor creates a removable hole at the excluded input and simplified output height. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Rational Functions and Holes: the useful representation: values

Case	Result
Key idea	A common factor creates a removable hole at the excluded input and simplified output height.
Boundary	$Q(a)\ne0$ after cancellation.

Use the structure in Rational Functions and Holes

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

A common factor creates a removable hole at the excluded input and simplified output height.
$Q(a)\ne0$ after cancellation.
Failure check: Erasing $x\ne a$ falsely fills the hole.

1.11

Equivalent Representations of Polynomial and Rational Expressions

What decides a correct solution for Equivalent Representations of Polynomial and Rational Expressions?

Short answer

Factored, expanded, and divided forms reveal different features of the same function.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Equivalent Representations of Polynomial and Rational Expressions

Use factored form for zeros and multiplicity, expanded form for coefficients and end behavior, and quotient-plus-remainder form for rational asymptotes.

Equivalent formulas must preserve restrictions inherited from the original expression. A useful representation is chosen for the feature being asked about. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

x^2-1=(x-1)(x+1)

Expansion and factoring preserve polynomial values.

Use when: For rational cancellation, original exclusions remain.
Do not assume: Treating a simplified rational formula as domain-equivalent can erase holes.

Equivalent Representations of Polynomial and Rational Expressions: the useful representation

Factored, expanded, and divided forms reveal different features of the same function. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Equivalent Representations of Polynomial and Rational Expressions: the useful representation: values

Case	Result
Key idea	Factored, expanded, and divided forms reveal different features of the same function.
Boundary	For rational cancellation, original exclusions remain.

Use the structure in Equivalent Representations of Polynomial and Rational Expressions

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Factored, expanded, and divided forms reveal different features of the same function.
For rational cancellation, original exclusions remain.
Failure check: Treating a simplified rational formula as domain-equivalent can erase holes.

1.12

Transformations of Functions

What decides a correct solution for Transformations of Functions?

Short answer

Transformations act on coordinates in a predictable inside-versus-outside order.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Transformations of Functions

Outside changes move or scale outputs; inside changes act oppositely on inputs. Track a landmark point and the domain rather than relying on memorized direction words.

A negative scale may reflect and reverse orientation. Composed transformations should be applied in coordinate order to avoid sign errors. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

g(x)=a f(b(x-h))+k

The parameters control vertical and horizontal scale, reflection, and translation.

Use when: $a,b\ne0$ and transformed inputs remain in the base domain.
Do not assume: Moving right for $f(x+h)$ reverses the actual horizontal shift.

Transformations of Functions: the useful representation

Transformations act on coordinates in a predictable inside-versus-outside order. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Transformations of Functions: the useful representation: values

Case	Result
Key idea	Transformations act on coordinates in a predictable inside-versus-outside order.
Boundary	$a,b\ne0$ and transformed inputs remain in the base domain.

Use the structure in Transformations of Functions

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Transformations act on coordinates in a predictable inside-versus-outside order.
$a,b\ne0$ and transformed inputs remain in the base domain.
Failure check: Moving right for $f(x+h)$ reverses the actual horizontal shift.

1.13

Function Model Selection and Assumption Articulation

What decides a correct solution for Function Model Selection and Assumption Articulation?

Short answer

Choose a model from the pattern of change and state why its assumptions fit the context.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Function Model Selection and Assumption Articulation

Constant differences support linear structure, constant second differences support quadratic structure, and stable ratios support exponential structure. Residuals reveal systematic mismatch.

A high numerical fit does not justify extrapolation beyond the observed domain. Identify relevant constraints, omitted variables, and whether the rate pattern is plausibly stable. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

\text{residual}=y-\widehat y

Residual sign and pattern evaluate model fit.

Use when: Data are measured on a meaningful domain with comparable units.
Do not assume: Choosing the largest $R^2$ without inspecting structure or context can select a misleading model.

Function Model Selection and Assumption Articulation: the useful representation

Choose a model from the pattern of change and state why its assumptions fit the context. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Function Model Selection and Assumption Articulation: the useful representation: values

Case	Result
Key idea	Choose a model from the pattern of change and state why its assumptions fit the context.
Boundary	Data are measured on a meaningful domain with comparable units.

Use the structure in Function Model Selection and Assumption Articulation

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Choose a model from the pattern of change and state why its assumptions fit the context.
Data are measured on a meaningful domain with comparable units.
Failure check: Choosing the largest $R^2$ without inspecting structure or context can select a misleading model.

1.14

Function Model Construction and Application

What decides a correct solution for Function Model Construction and Application?

Short answer

Construct a function from defining features, then use it only within its justified domain.

Know firstFunction notation · Algebraic structure · Graph interpretation

The decisive structure in Function Model Construction and Application

Translate zeros, asymptotes, rates, or anchor points into parameters and solve for remaining constants. Verify the result in a second representation before interpreting predictions.

Units and realistic input bounds belong to the model. A mathematically valid output may still be impossible in context. The figure isolates that relationship so the symbolic claim, relevant graph feature, and limiting condition can be checked against one another.

Formula

Formula and conditions

f(x)=a\prod_i(x-r_i)^{m_i}

Zeros and multiplicities build polynomial structure.

Use when: Enough independent conditions are supplied to determine the parameters.
Do not assume: Matching one data point does not determine an entire model family.

Function Model Construction and Application: the useful representation

Construct a function from defining features, then use it only within its justified domain. The display emphasizes the feature that carries the conclusion.

The interactive version of this figure is part of the full study guide.

Function Model Construction and Application: the useful representation: values

Case	Result
Key idea	Construct a function from defining features, then use it only within its justified domain.
Boundary	Enough independent conditions are supplied to determine the parameters.

Use the structure in Function Model Construction and Application

Identify the requested quantity and its domain.
Match the expression or graph to the defining relationship.
Apply the relationship while preserving its hypotheses.
Check sign, endpoints, units, or long-run behavior before concluding.

Keep these three ideas

Construct a function from defining features, then use it only within its justified domain.
Enough independent conditions are supplied to determine the parameters.
Failure check: Matching one data point does not determine an entire model family.

Cumulative review

What should be ready after this unit

This is a study map, not a quiz. Explain each item aloud, then return to the linked topic when the reasoning is incomplete.

Recognize immediately

Function family
Rate pattern

Reconstruct

Defining relation
Representation change

State or verify

Domain and range
Model assumptions

Read from a graph

Intercepts and restrictions
Long-run or periodic behavior

Catch in your own work

Does the representation preserve the domain?
Did I interpret each parameter?

Expect to see again

Calculus models
Data and geometric reasoning