Review AP Pre-Calculus Unit 1 to build fluency with polynomial and rational functions, from describing how quantities change in tandem to modeling real-world scenarios with regression and transformations. This unit carries 30-40% of the exam weight, making it the single most important unit to know well.
Use the topic guides, key terms, and practice questions available here to work through all 14 topics before your exam.
Unit 1 is the foundation of AP Pre-Calculus. It asks you to think about functions in four representations: graphical, numerical, analytical, and verbal. You will move fluidly between these representations to describe behavior, identify key features, and build models.
Topics 1.1-1.3 establish how input and output values vary together, how to compute average rates of change using (f(b)-f(a))/(b-a), and how those rates behave differently for linear functions (constant rate) versus quadratic functions (linearly changing rate).
Topics 1.4-1.6 focus on the form p(x) = a_n x^n + ... + a_0. You identify local and global extrema, analyze zeros and their multiplicities, determine whether the graph crosses or touches the x-axis, and use the degree and sign of the leading term to describe end behavior.
Topics 1.7-1.11 cover rational functions as quotients of polynomials. You compare degrees to find horizontal or slant asymptotes, locate zeros from the numerator, identify vertical asymptotes where the denominator is zero, and distinguish holes (removable discontinuities) from vertical asymptotes using multiplicity.
Topics 1.12-1.14 bring everything together. Transformations of the form g(x) = af(b(x+h)) + k let you build new functions from parent functions. Model selection depends on the pattern in the data: constant rates suggest linear, linearly changing rates suggest quadratic, and multiple turning points suggest higher-degree polynomials. Rational functions model inversely proportional relationships. You must also articulate the assumptions and domain or range restrictions that make a model valid in context.
Describe how input and output values vary together. Identify increasing and decreasing intervals, concavity, and zeros. Construct graphs from verbal descriptions of real situations.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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Review Rational Functions and End Behavior with attention to how the concept appears in AP-style source and evidence questions.
Review Polynomial Functions and Complex Zeros with attention to how the concept appears in AP-style source and evidence questions.
Review Rates of Change in Linear and Quadratic Functions with attention to how the concept appears in AP-style source and evidence questions.
Review Rational Functions and Zeros with attention to how the concept appears in AP-style source and evidence questions.
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.
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.
| 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 |
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.
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.
| 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 |
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.
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.
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.
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.
Try AP-style multiple-choice questions and written prompts after you review the notes.
| 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. |
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.
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.
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.
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.
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.
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.
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.
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.
Open the individual guides for Unit 1 when you want a closer review of one topic.
browse guidesPractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsWatch past review streams filtered to Unit 1 when you want a video walkthrough.
open videosUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Pre-Calc Unit 1 covers 14 topics across polynomial and rational functions, including rates of change, polynomial end behavior, complex zeros, rational functions and vertical asymptotes, rational functions and holes, transformations of functions, and function model construction. The unit opens with Change in Tandem (1.1) and builds through function model application (1.14). Here's a quick breakdown of the topic groups: - **Rates of change:** Topics 1.1-1.3 cover change in tandem, general rates of change, and rates of change in linear and quadratic functions. - **Polynomial functions:** Topics 1.4-1.6 cover polynomial rates of change, complex zeros, and end behavior. - **Rational functions:** Topics 1.7-1.10 cover end behavior, zeros, vertical asymptotes, and holes. - **Modeling and transformations:** Topics 1.11-1.14 cover equivalent expressions, transformations of functions, model selection, and real-world application. See AP Pre-Calc Unit 1 for matched practice on all 14 topics.
AP Pre-Calc Unit 1 makes up 30-40% of the AP exam, making it the heaviest-weighted unit on the test. That means roughly one-third or more of your exam score comes from polynomial functions, rational functions, rates of change, and transformations of functions. Prioritizing this unit pays off more than any other.
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.
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.
The best place to find AP Pre-Calc Unit 1 practice questions, including multiple-choice and practice test sets, is AP Pre-Calc Unit 1. That page organizes practice by all 14 topics, so you can target polynomial functions, rational functions, rates of change, or transformations of functions individually before taking a full unit practice test. For MCQ practice, focus on topics that appear most on the exam: end behavior (1.6, 1.7), zeros and asymptotes (1.8, 1.9), and transformations (1.12). For a practice test experience, work through all 14 topics in order to simulate the full unit's 30-40% exam weight.
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.