Change in Tandem
Describe how input and output values vary together. Identify increasing and decreasing intervals, concavity, and zeros. Construct graphs from verbal descriptions of real situations.
AP Precalculus Unit 1 Review: Polynomial and Rational Functions
- 30–40% of AP multiple choice
- 14 topics
- AP exam review verified for 2027
Unit 1 covers polynomial and rational functions, including rates of change, zeros, end behavior, asymptotes, holes, transformations, and function modeling. It accounts for 30-40% of the AP Pre-Calculus exam.
Fiveable's AP Precalculus Unit 1 review covers 14 topics, with guided practice questions for the unit.
AP Precalculus unit 1 topics
Rates of Change
Compute average rate of change as (f(b) - f(a)) / (b - a). Approximate the rate of change at a point using small intervals. Compare rates at two points and interpret positive and negative rates.
Rates of Change in Linear and Quadratic Functions
Linear functions have constant average rates of change. Quadratic functions have average rates that change at a constant rate over equal-length intervals. Connect these patterns to concavity.
Polynomial Functions and Rates of Change
Identify the degree, leading coefficient, and leading term of a polynomial. Locate local and global extrema and points of inflection. Connect increasing/decreasing behavior to rates of change.
Polynomial Functions and Complex Zeros
Find real and complex zeros. Use multiplicity to determine whether the graph crosses or touches the x-axis. Apply the conjugate pair rule and identify even and odd polynomial functions.
Polynomial Functions and End Behavior
Use the degree and sign of the leading term to determine end behavior. Write end behavior using limit notation. Distinguish even-degree from odd-degree end behavior patterns.
Rational Functions and End Behavior
Compare degrees of numerator and denominator to determine horizontal or slant asymptotes. Use the quotient of leading terms to describe end behavior. Apply limit notation for rational functions.
Rational Functions and Zeros
Find zeros of a rational function from the numerator. Use zeros and vertical asymptotes as partition points to solve rational inequalities with a sign chart.
Rational Functions and Vertical Asymptotes
Identify vertical asymptotes where the denominator is zero and the numerator is not. Use one-sided limits to describe behavior near a vertical asymptote. Compare multiplicities to distinguish asymptotes from holes.
Rational Functions and Holes
Identify holes where a common factor cancels from numerator and denominator. Find the coordinates of a hole by evaluating the simplified function at the excluded input value using a limit.
Equivalent Representations of Polynomial and Rational Expressions
Switch between factored and standard forms to extract different information. Use polynomial long division to rewrite rational expressions and find slant asymptotes. Expand binomials using Pascal's Triangle and the binomial theorem.
Transformations of Functions
Apply vertical and horizontal translations, dilations, and reflections to a parent function. Read the transformation parameters from g(x) = af(b(x + h)) + k. Track how domain and range change under each transformation.
Function Model Selection and Assumption Articulation
Choose linear, quadratic, polynomial, or rational models based on data patterns and context. State assumptions about what is constant and articulate domain and range restrictions that make the model valid.
Function Model Construction and Application
Build polynomial and rational models using transformations, regression, or interpolation. Use a model to predict output values, rates of change, and average rates of change. Construct piecewise models when data shows distinct regional behavior.
Practice AP Precalculus unit 1 questions
Try AP-style questions on unit 1, then use the review notes below for anything you miss.
AP Precalculus unit 1 practice test
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Unit 1 review notes
1.1
Functions and change in tandem
A function maps each input value to exactly one output value. The domain is the set of inputs and the range is the set of outputs. When you describe how a function behaves, you identify intervals where it is increasing or decreasing and whether the rate of change itself is increasing (concave up) or decreasing (concave down). Zeros occur where the output equals zero, which corresponds to x-intercepts on the graph.
- Increasing interval: For all a < b in the interval, f(a) < f(b); output rises as input rises.
- Decreasing interval: For all a < b in the interval, f(a) > f(b); output falls as input rises.
- Concave up: The rate of change is increasing over the interval; the graph curves upward.
- Concave down: The rate of change is decreasing over the interval; the graph curves downward.
- Zero of a function: An input value where f(x) = 0; corresponds to an x-intercept on the graph.
Given a verbal description of a situation, can you sketch a graph that correctly shows increasing/decreasing behavior, concavity, and zeros?
1.2
Average rates of change for linear and quadratic functions
The average rate of change over [a, b] equals (f(b) - f(a)) / (b - a), which is the slope of the secant line through (a, f(a)) and (b, f(b)). To approximate the rate of change at a single point, compute average rates over smaller and smaller intervals containing that point. For linear functions, the average rate of change is constant over any interval. For quadratic functions, average rates of change over consecutive equal-length intervals form a linear pattern, meaning those rates change at a constant rate.
- Average rate of change: (f(b) - f(a)) / (b - a); the slope of the secant line from (a, f(a)) to (b, f(b)).
- Linear function rate: Constant average rate of change over any interval; first differences are equal.
- Quadratic function rate: Average rates of change over equal-length intervals follow a linear pattern; second differences are constant.
- Positive rate of change: Both quantities increase or both decrease together.
- Negative rate of change: As one quantity increases, the other decreases.
Given a table of values, can you compute average rates of change and determine whether the data is better modeled by a linear or quadratic function based on the pattern of those rates?
| Function type | Average rate of change pattern | Concavity signal |
|---|---|---|
| Linear | Constant (first differences equal) | Neither concave up nor down |
| Quadratic (a > 0) | Linearly increasing (second differences positive) | Concave up |
| Quadratic (a < 0) | Linearly decreasing (second differences negative) | Concave down |
1.4
Polynomial structure: extrema, zeros, and multiplicity
A polynomial p(x) = a_n x^n + ... + a_0 has degree n and leading coefficient a_n. Where the function switches from increasing to decreasing, it has a local maximum; where it switches from decreasing to increasing, it has a local minimum. A degree-n polynomial has at most n-1 turning points. Every real zero a corresponds to a linear factor (x - a). The multiplicity of a zero determines graph behavior at the x-intercept: odd multiplicity means the graph crosses the x-axis; even multiplicity means it touches and turns back. A degree-n polynomial has exactly n complex zeros counting multiplicities, and non-real zeros come in conjugate pairs.
- Local maximum: Output is greater than at nearby inputs; function switches from increasing to decreasing.
- Global maximum: The greatest output value among all local maxima over the entire domain.
- Multiplicity: The number of times a factor (x - a) appears; odd multiplicity crosses the x-axis, even multiplicity touches it.
- Conjugate pair: If a + bi is a non-real zero of a polynomial with real coefficients, then a - bi is also a zero.
- Even/odd function: Even: f(-x) = f(x), symmetric over x = 0. Odd: f(-x) = -f(x), symmetric about the origin.
Given p(x) = (x - 2)^2 (x + 3), identify the zeros, their multiplicities, and describe the graph behavior at each x-intercept.
1.6
End behavior of polynomial and rational functions
For polynomials, end behavior is determined entirely by the leading term a_n x^n. Even-degree polynomials have both ends going the same direction; odd-degree polynomials have ends going in opposite directions. The sign of a_n determines whether the right end goes up or down. For rational functions, compare the degrees of the numerator and denominator. If the denominator dominates, the end behavior approaches y = 0. If the degrees are equal, the end behavior approaches y = (leading coefficient of numerator) / (leading coefficient of denominator). If the numerator dominates by exactly one degree, there is a slant asymptote found by polynomial long division.
- Leading term dominance: For large |x|, the leading term a_n x^n determines the output of a polynomial.
- Horizontal asymptote y = 0: Occurs when the degree of the denominator exceeds the degree of the numerator.
- Horizontal asymptote y = a_n/b_m: Occurs when numerator and denominator have equal degree; ratio of leading coefficients.
- Slant asymptote: Occurs when the numerator degree is exactly one more than the denominator degree; found by polynomial long division.
- Limit notation: lim_{x→∞} p(x) = ∞ means output increases without bound as input increases without bound.
For r(x) = (3x^3 - x) / (x^2 + 2), determine the end behavior and identify whether there is a horizontal or slant asymptote.
| Degree comparison (num vs denom) | End behavior type | Asymptote |
|---|---|---|
| Denominator degree greater | Approaches zero | Horizontal: y = 0 |
| Degrees equal | Approaches constant | Horizontal: y = leading coeff ratio |
| Numerator degree one greater | Grows without bound (linear) | Slant asymptote |
| Numerator degree two or more greater | Grows without bound (polynomial) | No horizontal or slant asymptote |
1.8
Rational function zeros, vertical asymptotes, and holes
The real zeros of a rational function come from the real zeros of the numerator that are in the domain. A vertical asymptote at x = a occurs when a is a zero of the denominator but not the numerator, or when its multiplicity in the denominator exceeds its multiplicity in the numerator. Near a vertical asymptote, one-sided limits go to positive or negative infinity. A hole (removable discontinuity) occurs at x = c when the same factor (x - c) cancels from both numerator and denominator and the multiplicity in the numerator is greater than or equal to its multiplicity in the denominator. The y-coordinate of the hole is found by evaluating the simplified function at x = c.
- Zero of a rational function: A real zero of the numerator that is in the domain of the rational function.
- Vertical asymptote: x = a where a is a zero of the denominator with higher multiplicity than in the numerator; one-sided limits go to infinity.
- Removable discontinuity (hole): At x = c where a common factor cancels; the graph has an open circle at (c, L) where L is the limit.
- One-sided limit: lim_{x→a^+} or lim_{x→a^-}; describes behavior approaching a from the right or left.
- Sign chart: A number-line analysis using zeros and vertical asymptotes as partition points to determine where r(x) > 0 or r(x) < 0.
For r(x) = (x^2 - 4) / (x^2 - x - 2), factor completely, then identify all zeros, vertical asymptotes, and holes with their coordinates.
1.11
Equivalent representations and polynomial long division
Factored form reveals zeros, x-intercepts, holes, and asymptotes. Standard form reveals end behavior through the leading term. Polynomial long division rewrites f(x) / g(x) as q(x) + r(x)/g(x), where the degree of r is less than the degree of g. This is essential for finding slant asymptotes and for simplifying rational expressions. The binomial theorem uses Pascal's Triangle to expand (a + b)^n without repeated multiplication.
- Factored form: Written as a product of factors; reveals zeros, x-intercepts, holes, and asymptotes directly.
- Standard form: Written in descending powers; reveals degree and leading coefficient for end behavior.
- Polynomial long division: Divides f(x) by g(x) to get quotient q(x) plus remainder r(x)/g(x); used to find slant asymptotes.
- Binomial theorem: Uses Pascal's Triangle entries to expand (a + b)^n; the nth row gives the coefficients for the expansion.
- Pascal's Triangle: A triangular array where each entry is the sum of the two above it; row n gives binomial coefficients for (a + b)^n.
Divide (2x^3 + x^2 - 3x + 1) by (x - 1) using polynomial long division and interpret the result in terms of the graph of the corresponding rational function.
1.12
Transformations of functions
Starting from a parent function f(x), you can build g(x) = af(b(x + h)) + k. Adding k shifts the graph up or down (vertical translation). Replacing x with (x + h) shifts the graph left or right by -h units (horizontal translation). Multiplying by a stretches or compresses vertically by factor |a| and reflects over the x-axis if a < 0. Replacing x with bx compresses or stretches horizontally by factor 1/|b| and reflects over the y-axis if b < 0. Transformations affect the domain and range of the function.
- Vertical translation: g(x) = f(x) + k; shifts the graph up k units (k > 0) or down |k| units (k < 0).
- Horizontal translation: g(x) = f(x + h); shifts the graph left h units (h > 0) or right |h| units (h < 0).
- Vertical dilation: g(x) = af(x); stretches by |a| if |a| > 1, compresses if 0 < |a| < 1, reflects over x-axis if a < 0.
- Horizontal dilation: g(x) = f(bx); compresses by factor 1/|b| if |b| > 1, stretches if 0 < |b| < 1, reflects over y-axis if b < 0.
- Multiplicative transformation: A transformation involving multiplication by a constant, producing dilation or reflection.
Given g(x) = -2f(3(x - 1)) + 4, describe each transformation applied to f in order and state how the domain and range change.
1.13
Function model selection, construction, and application
Choosing the right model type depends on the pattern in the data. Constant first differences indicate a linear model; constant second differences indicate a quadratic model; constant nth differences indicate a degree-n polynomial model. Geometric contexts with area often use quadratic models; volume contexts often use cubic models. Rational functions model inversely proportional relationships such as gravitational or electromagnetic force. Once a model type is selected, you construct it using transformations of a parent function, regression on a calculator, or interpolation. You must also state assumptions (what is held constant, how quantities change) and domain or range restrictions (nonnegative time, integer outputs) that make the model valid.
- nth differences: Successive differences computed n times; constant nonzero nth differences indicate a degree-n polynomial model.
- Regression: A calculator-based technique that fits a polynomial model (linear, quadratic, cubic, quartic) to a data set.
- Inversely proportional: y = k/x or y = k/x^2; modeled by rational functions; as one quantity increases, the other decreases.
- Piecewise-defined function: A function using different expressions on different intervals; used when data shows distinct behavior in different regions.
- Domain restriction: A limitation on input values based on context (e.g., time cannot be negative) or mathematical constraints.
A table of data shows that second differences are approximately constant and equal to 4. What type of function should you use to model the data, and what does the constant second difference tell you about the leading coefficient?
Key terms
| Term | Definition |
|---|---|
| leading term | The term a_n x^n in a polynomial with the highest degree; determines end behavior and dominates for large |x|. |
| multiplicity | The number of times a factor (x - a) appears in a polynomial's factored form; odd multiplicity means the graph crosses the x-axis at x = a, even multiplicity means it touches and turns back. |
| real zero | A real number a where p(a) = 0; corresponds to a linear factor (x - a) and an x-intercept at (a, 0) on the graph. |
| local maximum | A point where a polynomial switches from increasing to decreasing; output is greater than at nearby input values. |
| global maximum | The greatest output value among all local maxima over the entire domain of the function; also called the absolute maximum. |
| horizontal asymptote | A horizontal line y = b that a rational function approaches as x increases or decreases without bound; determined by comparing degrees of numerator and denominator. |
| removable discontinuity | A hole in the graph of a rational function at x = c where a common factor cancels; the graph approaches a finite value L at that point, written as an open circle at (c, L). |
| one-sided limit | The value a function approaches as input approaches a from the right (lim_{x→a^+}) or from the left (lim_{x→a^-}); used to describe behavior near vertical asymptotes. |
| factored form | A polynomial or rational expression written as a product of its factors; directly reveals zeros, x-intercepts, holes, and asymptotes. |
| standard form | A polynomial written in descending powers as a_n x^n + ... + a_0; reveals degree and leading coefficient for end behavior analysis. |
| nth differences | Successive differences computed n times from a data table; constant nonzero nth differences indicate the data fits a degree-n polynomial model. |
| regression | A calculator-based technique that fits a polynomial model (linear, quadratic, cubic, or quartic) to a data set by finding the best-fitting curve. |
| piecewise-defined function | A function defined by different expressions on different intervals; used when data or context shows distinct behavior in separate regions. |
| binomial theorem | Uses entries from a row of Pascal's Triangle to expand (a + b)^n without repeated multiplication; the nth row gives the coefficients for the expansion. |
How to review unit 1
Topics students miss most
Miss rates come from Fiveable multiple-choice practice.
Mistakes to avoid in Polynomial and Rational Functions
Confusing holes with vertical asymptotes
A hole occurs when a factor cancels completely (numerator multiplicity is greater than or equal to denominator multiplicity). A vertical asymptote occurs when the denominator zero does not cancel. Always factor and compare multiplicities before labeling a discontinuity.
Misreading horizontal translation direction
In g(x) = f(x + h), the graph shifts left by h units when h is positive, not right. Students frequently reverse this. Remember: replacing x with (x + 3) moves the graph 3 units to the left.
Applying end behavior rules without checking degree parity
Even-degree polynomials have both ends going the same direction; odd-degree polynomials have ends going in opposite directions. Mixing these up leads to incorrect end behavior descriptions, especially for degree-4 versus degree-3 functions.
Forgetting that even multiplicity zeros touch but do not cross
At a zero with even multiplicity, the graph touches the x-axis and turns back. At odd multiplicity, it crosses. Sketching a graph that crosses at every zero is a frequent error when multiplicities are not checked.
Choosing the wrong model type from a data table
Check differences systematically: first differences constant means linear, second differences constant means quadratic, third differences constant means cubic. Jumping to a model type without computing differences leads to poor model selection on contextual problems.
Polynomial and Rational Functions on the AP exam
Reading functions across multiple representations
AP Pre-Calculus consistently asks you to extract information from a function given in graphical, numerical, analytical, or verbal form. For Unit 1, this means identifying end behavior from an equation, reading zeros and asymptotes from a graph, computing average rates of change from a table, and translating a verbal description into a sketch. Practice moving between all four representations for both polynomial and rational functions.
Justifying conclusions about function behavior
Free-response tasks in AP Pre-Calculus require written justification, not just numerical answers. For Unit 1, you should be able to explain why a rational function has a hole rather than a vertical asymptote by citing multiplicity, explain why a polynomial's end behavior follows from its leading term, and justify a model choice by referencing the pattern of differences in a data table. Use precise vocabulary: multiplicity, leading coefficient, concavity, and limit notation.
Contextual modeling with domain and range restrictions
Unit 1 modeling tasks present real-world scenarios where you must select a function type, construct the model, and answer questions about predicted values or rates of change. Exam tasks often require you to state assumptions explicitly and apply domain or range restrictions based on context, such as restricting time to nonnegative values or rounding outputs to whole numbers. Rational function models appear in contexts involving inversely proportional quantities.
Study unit 1 other ways
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Unit 1 printables
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get printablesFrequently Asked Questions
What's on the AP Pre-Calc Unit 1 progress check (MCQ and FRQ)?
The AP Pre-Calc Unit 1 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all 14 topics in the unit. MCQ questions test your ability to interpret polynomial and rational functions, identify end behavior, locate zeros and vertical asymptotes, and analyze rates of change. FRQ questions typically ask you to construct or analyze a function model, justify your reasoning about transformations of functions, or interpret change in tandem from a graph or table. The progress check pulls heavily from these topic clusters: - Rates of change (1.1-1.3)
- Polynomial functions, zeros, and end behavior (1.4-1.6)
- Rational functions, asymptotes, and holes (1.7-1.10)
- Transformations and function modeling (1.11-1.14) Practicing with questions matched to each topic before you take the progress check is the most efficient prep. Visit AP Pre-Calc Unit 1 for that practice.
How do I practice AP Pre-Calc Unit 1 FRQs?
AP Pre-Calc Unit 1 FRQs most often come from function modeling and transformations of functions, specifically Topics 1.12-1.14, where you construct a model, state assumptions, and interpret outputs. You'll also see FRQ-style questions built around rates of change in polynomial functions and analyzing rational functions with asymptotes or holes. To practice effectively: 1. Know what the question is asking. Unit 1 FRQs usually ask you to select a function type, justify why it fits the data, and apply it. Practice articulating your reasoning in writing, not just computing answers.
2. Work through Topics 1.13 and 1.14 closely. Function Model Selection and Function Model Construction are the most FRQ-heavy topics in this unit.
3. Check your end behavior and zeros work. FRQs on polynomial and rational functions often include a part that asks you to describe or justify end behavior.
4. Use past AP Classroom FRQ prompts alongside topic-level practice at AP Pre-Calc Unit 1.
How should I study AP Pre-Calc Unit 1?
Start with rates of change (Topics 1.1-1.3) because understanding how functions change sets up everything else in Unit 1. From there, build your understanding of polynomial functions and rational functions in sequence, since end behavior, zeros, and asymptotes each build on the previous topic. Here's a concrete study plan: 1. Topics 1.1-1.3 first. Nail change in tandem and rates of change before moving on. These ideas show up throughout the unit.
2. Work polynomial functions as a block (1.4-1.6). Focus on connecting complex zeros to the graph and understanding end behavior rules.
3. Then tackle rational functions (1.7-1.10). Vertical asymptotes, holes, zeros, and end behavior are all tested heavily. Practice sketching graphs from equations.
4. Finish with transformations and modeling (1.11-1.14). Transformations of functions and function model construction are the most likely FRQ topics, so spend real time here.
5. Practice by topic, then by unit. Use AP Pre-Calc Unit 1 to check your understanding topic by topic before doing a full unit review. Since Unit 1 is 30-40% of the exam, returning to it during your final review is worth the time.