AP Pre-Calculus Unit 1 ReviewPolynomial and Rational Functions

AP Precalculus Unit 1 is about how two quantities change together, and it uses polynomial and rational functions as the lab for studying that change. The single biggest idea is rate of change: every graph feature you analyze in this unit (zeros, asymptotes, end behavior, concavity) is really a statement about how outputs respond as inputs change. At 30-40% of the exam, this is the heaviest unit in the course, and the function-analysis habits you build here get reused in every unit after it.

What this unit covers

Covariation and rates of change (Topics 1.1-1.3)

• A function pairs each input with exactly one output. The unit starts by asking how those values vary in tandem, meaning where a function is increasing or decreasing and whether it does so faster or slower over time.
• The average rate of change over [a, b] is the slope of the secant line through (a, f(a)) and (b, f(b)). It is the constant rate that would produce the same total change over the interval.
• Concavity is about the rate of change of the rate of change. Concave up means the rate of change is increasing; concave down means it is decreasing. You should be able to read this off a graph and explain it in a sentence.
• Each function family has a signature rate-of-change pattern. Linear functions have a constant rate of change. Quadratic functions have a rate of change that itself changes at a constant rate (the average rates of change over equal-length intervals form a linear pattern). This "differences of differences" idea is how you identify function types from tables.

Polynomial behavior: zeros, multiplicity, end behavior (Topics 1.4-1.6)

• A degree-n polynomial p(x) = a_n x^n + ... + a_1 x + a_0 has exactly n complex zeros when you count multiplicity. If a is a real zero, then (x - a) is a linear factor and x = a is an x-intercept.
• Multiplicity controls graph behavior at a zero. Odd multiplicity means the graph crosses the x-axis there; even multiplicity means it touches and turns around.
• Between every two distinct real zeros there is at least one local maximum or minimum. A polynomial of degree n has at most n - 1 local extrema, and points of inflection mark where concavity switches.
• End behavior is decided entirely by the leading term. You express it with limit notation, like lim as x approaches infinity of p(x) equals infinity. Even degree means both ends go the same direction; odd degree means opposite directions; the sign of the leading coefficient flips everything.
• Even functions satisfy f(-x) = f(x) and have symmetry across x = 0. Odd functions satisfy f(-x) = -f(x) and have symmetry about the origin.

Rational functions: zeros, asymptotes, and holes (Topics 1.7-1.10)

• A rational function is a quotient of two polynomials, and its whole personality comes from comparing the numerator and denominator. Real zeros come from the numerator (as long as the value is in the domain).
• End behavior is a degree contest. If the denominator's degree wins, the horizontal asymptote is y = 0. If the degrees tie, the horizontal asymptote is the ratio of leading coefficients. If the numerator's degree is exactly one more, there is a slant asymptote you find by long division.
• A zero of the denominator that is not canceled gives a vertical asymptote. Near a vertical asymptote, output values increase or decrease without bound, written with limits like lim as x approaches a from the left.
• If a factor's multiplicity in the numerator is greater than or equal to its multiplicity in the denominator, the graph has a hole instead of an asymptote at that input. You find the hole's y-value by checking outputs for inputs close to that x-value.
• Real zeros of the numerator and denominator are the boundary points when you solve rational inequalities like r(x) ≥ 0 with a sign chart.

Equivalent forms and transformations (Topics 1.11-1.12)

• Different algebraic forms reveal different things. Factored form shows zeros, holes, and asymptotes. Standard form shows degree, leading coefficient, and end behavior. Choosing the right form is half the battle on free-response questions.
• Polynomial long division rewrites f(x) as g(x)q(x) + r(x), where the remainder's degree is less than the divisor's. This is exactly how you find slant asymptote equations.
• The binomial theorem uses a row of Pascal's Triangle to expand (a + b)^n without multiplying everything out by hand.
• Transformations build new functions from parent functions. g(x) = f(x) + k shifts vertically, g(x) = f(x + h) shifts horizontally by -h, g(x) = a f(x) dilates vertically, and g(x) = f(bx) dilates horizontally by a factor of 1/b.

Modeling with function models (Topics 1.13-1.14)

• Picking a model means matching the pattern in the data. Roughly constant rate of change suggests linear. Roughly linear rate of change, or symmetric data with one max or min, suggests quadratic. Area contexts tend to be quadratic; volume contexts tend to be cubic.
• Inverse proportionality calls for a rational model. Gravitational and electromagnetic force are both inversely proportional to squared distance, which is why physics keeps showing up in these problems.
• Every model carries assumptions and restrictions. You may need to restrict the domain (a side length cannot be negative) or the range (you cannot sell half a ticket) based on context.
• You build models with transformations of parent functions, with piecewise definitions, or with technology using linear, quadratic, cubic, or quartic regression, then use the model to predict values and rates of change with correct units.

Unit 1, Polynomial and Rational Functions at a glance

Why Unit 1, Polynomial and Rational Functions matters in AP Pre-Calc

AP Precalculus is organized around a few recurring ideas, and Unit 1 introduces all of them. The course keeps asking the same three questions about every function family: how does it change, what does its graph look like and why, and what real situations does it model? Unit 1 answers those questions for polynomials and rationals, and the vocabulary you build here (rate of change, concavity, end behavior, zeros, transformations) is the vocabulary of the whole course.

• Rate of change is the through-line of the course and the on-ramp to calculus. Average rate of change in this unit becomes the derivative idea later in your math career.
• Transformations learned here are the universal toolkit. Every later function family gets shifted, stretched, and reflected with the exact same rules.
• The modeling cycle (pick a function type, justify it, restrict the domain, interpret with units) is repeated with exponentials, trig functions, and beyond.

How this unit connects across the course

• Exponential and logarithmic functions (Unit 2) are defined by contrast with this unit. Polynomials have rates of change that change at predictable polynomial rates, while exponentials grow proportionally to their value. You need Unit 1's rate-of-change language to even state that difference, and Unit 2 reuses average rate of change over equal-length intervals to identify exponential patterns in tables.
• Trigonometric and polar functions (Unit 3) lean hard on transformations. A sinusoid like a sin(b(x + c)) + d is just the Unit 1 transformation rules applied to sin(x), with amplitude, period, and midline playing the roles of dilations and translations.
• Functions involving parameters, vectors, and matrices (Unit 4) extends the modeling work from Topics 1.13-1.14. Parametric functions describe quantities changing in tandem, which is literally the Topic 1.1 idea with a third variable driving both.
• Limit notation introduced for end behavior and vertical asymptotes here (Topics 1.6, 1.7, 1.9) is the same notation that opens AP Calculus, so getting fluent now pays off twice.

Key formulas and procedures

• Average rate of change: (f(b) - f(a)) / (b - a) over [a, b]. The slope of the secant line; use it to compare how fast a function changes on different intervals.
• Polynomial standard form: p(x) = a_n x^n + ... + a_1 x + a_0. The leading term a_n x^n alone determines end behavior.
• End behavior with limits: write statements like lim as x approaches infinity of p(x) = infinity. Required notation on the free-response section.
• Zeros and factors: a is a real zero of p if and only if (x - a) is a factor. A degree-n polynomial has exactly n complex zeros counting multiplicity.
• Multiplicity rule: odd multiplicity means the graph crosses the x-axis at that zero; even multiplicity means it is tangent to the axis and turns around.
• Even/odd tests: f(-x) = f(x) means even (symmetric across the y-axis); f(-x) = -f(x) means odd (symmetric about the origin).
• Vertical asymptote test: x = a is a vertical asymptote when a is a zero of the denominator with greater multiplicity than it has in the numerator.
• Hole test: a hole occurs at x = c when the numerator's multiplicity at c is greater than or equal to the denominator's. Estimate the hole's output from nearby input values.
• Horizontal/slant asymptotes: compare degrees. Denominator bigger gives y = 0; equal degrees give y = ratio of leading coefficients; numerator exactly one degree bigger gives a slant asymptote from long division.
• Polynomial long division: rewrite f(x) = g(x)q(x) + r(x) with deg(r) < deg(g). Use it to find slant asymptotes and equivalent forms.
• Binomial theorem: expand (a + b)^n using the entries in row n of Pascal's Triangle.
• Transformation rules: g(x) = a f(b(x + h)) + k shifts horizontally by -h, dilates vertically by a, dilates horizontally by 1/b, and shifts vertically by k.
• Model selection by rate of change: constant rate suggests linear, linearly changing rate suggests quadratic, inverse proportionality suggests rational. Cubic and quartic regressions handle wigglier data.

Unit 1, Polynomial and Rational Functions on the AP exam

This unit is 30-40% of the exam, the largest share of any unit, so polynomial and rational function skills show up everywhere on both the multiple-choice and free-response sections.

• Multiple-choice questions ask you to identify zeros, asymptotes, holes, and end behavior from equations, graphs, and tables, and to match a transformation to its effect on a graph. Some appear in the calculator-active portion, where regression and numerical zero-finding come into play.
• Free-response questions in this course follow set formats, and Unit 1 content anchors the function-concepts and modeling tasks. Expect to construct a model from a data table or scenario, justify your choice of function type using rate-of-change patterns, state domain restrictions, and interpret a value or an average rate of change in context with units.
• Justification language matters. "The graph crosses at x = 2 because the zero has odd multiplicity" or "the rate of change is increasing, so the graph is concave up" is the kind of sentence that earns points. Limit notation for end behavior and asymptote behavior is expected, not optional.
• Watch for equivalent-forms questions that hand you one form (say, standard form) and ask something the other form answers (zeros). Recognizing which form to convert to is itself the tested skill.

Essential questions

• What does a function's rate of change tell you about the shape of its graph, and how can you detect that from a table, a graph, or an equation?
• Why does the leading term control a polynomial's long-run behavior while the factors control its behavior near the x-axis?
• When does a rational function have a vertical asymptote versus a hole, and what is actually happening to output values at those inputs?
• How do you choose, justify, and restrict a function model so it makes sense for a real-world scenario?

Key terms to know

• Average rate of change: the change in output divided by the change in input over an interval, equal to the slope of the secant line.
• Concave up / concave down: the graph shape when the rate of change is increasing (up) or decreasing (down).
• Degree: the highest power of x in a polynomial, which sets the maximum number of zeros and controls end behavior.
• Leading coefficient: the coefficient of the highest-degree term; its sign flips end behavior direction.
• Zero (root): an input value a where p(a) = 0; real zeros correspond to x-intercepts.
• Multiplicity: the number of times a factor repeats; determines whether the graph crosses or touches at a zero.
• Point of inflection: a point where a graph changes concavity, meaning the rate of change switches from increasing to decreasing or vice versa.
• End behavior: what output values do as inputs increase or decrease without bound, written with limit notation.
• Rational function: a quotient of two polynomial functions, with behavior driven by comparing numerator and denominator.
• Vertical asymptote: a line x = a the graph approaches as outputs increase or decrease without bound, caused by an uncanceled denominator zero.
• Hole (removable discontinuity): a missing point at x = c where a common factor cancels, with numerator multiplicity at least matching the denominator's.
• Slant asymptote: the linear end-behavior line of a rational function whose numerator degree is exactly one more than the denominator's, found by long division.
• Binomial theorem: a shortcut for expanding (a + b)^n using Pascal's Triangle entries.
• Regression: a technology-based method for fitting a linear, quadratic, cubic, or quartic model to data.

Common mix-ups

• A zero of the denominator is not automatically a vertical asymptote. Factor first. If the factor cancels (numerator multiplicity is at least as big), you get a hole, not an asymptote.
• Multiplicity confusion costs points. Even multiplicity means the graph touches the x-axis and bounces; odd means it passes through. Do not assume every zero is a crossing.
• "Concave up" does not mean "increasing." A function can be decreasing and concave up (falling, but more and more slowly). Concavity is about the rate of change changing, not the direction of the function.
• Horizontal translations move opposite to intuition. g(x) = f(x + 3) shifts the graph 3 units left, not right, because the input reaches each value 3 units earlier.