Polynomial & Rational Functions

What this unit covers

The topics below follow the published Precalculus course framework for Unit 1. This unit is worth 30–40% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• End Behavior of Polynomials12 min · 3 objectivesPredict end behavior from the degree and leading coefficient of a polynomial · Explain why the leading term dominates for large |x| · Match a polynomial to the correct “arrows” description of its long-run behavior
• Zeros & Multiplicity13 min · 3 objectivesFind the real zeros of a factored polynomial · Use multiplicity to decide whether the graph crosses or touches the x-axis · Relate the degree to the total number of zeros counted with multiplicity
• Asymptotes of Rational Functions14 min · 3 objectivesLocate vertical asymptotes from the denominator of a rational function · Determine the horizontal asymptote by comparing degrees · Distinguish a hole (removable discontinuity) from a vertical asymptote
• Transformations of Functions13 min · 3 objectivesTranslate a graph horizontally and vertically from the equation · Identify reflections across the x-axis and y-axis · Combine shifts and reflections in the correct order
• Average Rate of Change14 min · 3 objectivesCompute the average rate of change of a function over an interval · Interpret an average rate of change as the slope of a secant line, with units · Explain why average rate of change over a shrinking interval approaches the slope at a point
• Rates of Change, Concavity & Inflection14 min · 3 objectivesDecide whether a rate of change is increasing or decreasing from a table or graph · Connect an increasing rate of change to concave-up behavior and a decreasing rate to concave-down · Locate a point of inflection as the place where concavity changes
• Polynomial Division & the Remainder Theorem14 min · 3 objectivesDivide one polynomial by another and write the result as quotient plus remainder over divisor · Use the Remainder Theorem to evaluate a polynomial and to test for factors · Use a known factor to reduce a polynomial and find its remaining zeros
• Complex Zeros & the Fundamental Theorem of Algebra14 min · 3 objectivesState how many zeros a polynomial of degree n has, counted with multiplicity · Use conjugate pairs to reconstruct a polynomial from partial information about its zeros · Explain why a real polynomial of odd degree must have a real zero
• Polynomial & Rational Inequalities14 min · 3 objectivesSolve a polynomial inequality using a sign chart built from its zeros · Solve a rational inequality without multiplying through by the denominator · Decide correctly whether each endpoint belongs in the solution set
• Holes, Asymptotes & Removable Discontinuities14 min · 3 objectivesDistinguish a hole from a vertical asymptote by factoring numerator and denominator · Find the coordinates of a hole, not just its x value · Explain why a graphing tool may not show a hole at all
• Slant Asymptotes & End-Behavior Models14 min · 3 objectivesPredict from the degrees alone whether a rational function has a horizontal or slant asymptote · Find a slant asymptote by polynomial division · Describe the end behavior of a rational function whose numerator degree exceeds the denominator by two or more
• Composition of Functions & Domain Restrictions14 min · 3 objectivesEvaluate and build compositions of two functions · Determine the domain of a composition from both inner and outer requirements · Decompose a given function into an inner and outer function
• Even, Odd & Symmetry14 min · 3 objectivesTest a function algebraically for even or odd symmetry · Connect even symmetry to reflection across the y-axis and odd symmetry to rotation about the origin · Predict the parity of a sum or product of functions of known parity
• Choosing and Validating a Function Model15 min · 3 objectivesSelect a model family from the pattern in a data table · Use residuals to judge whether a model fits, rather than relying on a correlation value · Explain why a better fit is not the same as a better model

Formulas in Unit 1

Every term in Unit 1

All 87 terms we publish for Polynomial & Rational Functions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Inverse function

Reflects the graph across y = x and swaps domain and range. Exists only if the original function is one-to-one, which the horizontal line test decides.

Average rate of change

[f(b) − f(a)]/(b − a) — the slope of the secant line through the two points. Precalculus's stand-in for the derivative.

Increasing vs decreasing rate of change

If the average rate of change is itself increasing, the graph is concave up; if decreasing, concave down. This is how concavity is described without calculus.

Degree and end behavior

Even degree with a positive leading coefficient rises on both ends; odd degree with positive leading coefficient falls left and rises right. The leading term alone decides it.

Zeros and multiplicity

A zero of odd multiplicity crosses the x-axis; a zero of even multiplicity touches and turns back. Multiplicity 3 or more crosses with a flattening.

Fundamental Theorem of Algebra

A degree-n polynomial has exactly n complex zeros counted with multiplicity. Non-real zeros of a real polynomial come in conjugate pairs.

Complex conjugate pairs

If a + bi is a zero of a polynomial with real coefficients, so is a − bi. This is why a real cubic always has at least one real zero.

Turning points

A polynomial of degree n has at most n − 1 turning points. Fewer is possible; more is not.

Polynomial long division

Divides p(x) by d(x) to give quotient plus remainder over divisor. The remainder's degree must be less than the divisor's.

Remainder Theorem

The remainder when p(x) is divided by (x − a) equals p(a). So p(a) = 0 exactly when (x − a) is a factor.

Rational function domain

All reals except where the denominator is zero. Those excluded values become either vertical asymptotes or holes.

Hole vs vertical asymptote

A factor canceling from both numerator and denominator leaves a hole; a factor remaining in the denominator alone gives a vertical asymptote.

Horizontal asymptote rules

Denominator degree larger: y = 0. Degrees equal: ratio of leading coefficients. Numerator degree larger: no horizontal asymptote.

Slant asymptote

Occurs when the numerator's degree is exactly one more than the denominator's. Found by polynomial division; it is the quotient, ignoring the remainder.

End behavior of a rational function

Determined by the ratio of leading terms, since lower-order terms become negligible as |x| grows.

Even and odd functions

Even: f(−x) = f(x), symmetric about the y-axis. Odd: f(−x) = −f(x), symmetric about the origin. Most functions are neither.

Function transformations

y = a·f(b(x − h)) + k. a scales vertically, b scales horizontally by 1/b, h shifts right, k shifts up. Horizontal changes act in reverse.

Composition of functions

(f∘g)(x) = f(g(x)). The domain requires x to be in g's domain AND g(x) to be in f's domain — the second condition is the one usually forgotten.

Piecewise function

Defined by different rules on different intervals. Continuity at a boundary requires the two pieces to agree in value there.

Extrema and intervals

An absolute extremum is the largest or smallest value on the whole domain; a relative extremum is largest or smallest only nearby.

Concavity and points of inflection

Concavity is the direction the graph bends; a point of inflection is where it changes. Rates of change switch from increasing to decreasing there.

Binomial expansion

The coefficients of (a + b)ⁿ are the binomial coefficients C(n, k), which are row n of Pascal's triangle.

Modeling with polynomial regression

A degree-n polynomial can pass exactly through n + 1 points, so higher degree always fits better — which is why fit alone does not justify a model.

Describing function behavior in words

The exam awards points for phrasing: state the interval, whether the function is increasing or decreasing, and whether it does so at an increasing or decreasing rate.

Interval notation

Parentheses exclude an endpoint, brackets include it. Infinity always takes a parenthesis, since it is not a number that can be attained.

Domain restrictions to check

Denominators cannot be zero, even roots need non-negative radicands, and logarithm arguments must be strictly positive.

Difference quotient

[f(x + h) − f(x)]/h — the average rate of change over an interval of width h, and the expression calculus takes a limit of.

Zeros, roots, x-intercepts and solutions

Four names for the same thing: the values where the function output is zero.

Sign chart

Mark every zero and undefined point on a number line, then test the sign in each interval. Solves inequalities without guessing.

Solving polynomial inequalities

Move everything to one side, factor, and read the sign chart. Never multiply both sides by a variable expression, whose sign is unknown.

Rational Root Theorem

Any rational zero of a polynomial with integer coefficients is ±(factor of the constant)/(factor of the leading coefficient) — a finite list to test.

Synthetic division

A shortcut for dividing by (x − a). The last entry is the remainder, which by the Remainder Theorem equals p(a).

Function notation and evaluation

f(x + 2) shifts the input, while f(x) + 2 shifts the output. They give different graphs and the confusion is common.

Symmetry of a rational function

Test f(−x): equal to f(x) means y-axis symmetry, equal to −f(x) means origin symmetry.

Piecewise continuity check

Evaluate both pieces at the boundary. If they disagree the graph jumps; if they agree it is continuous there.

Absolute value as a piecewise function

|x| equals x when x ≥ 0 and −x when x < 0. Rewriting this way is how absolute value equations and inequalities are solved reliably.

Solving absolute value inequalities

|u| < a means −a < u < a; |u| > a means u < −a or u > a. The second gives two separate intervals, not one.

Secant line vs tangent line

A secant passes through two points on a curve and its slope is the average rate of change. A tangent touches at one point and its slope is the instantaneous rate. Shrinking the interval turns the first into the second.

Units on an average rate of change

(units of output) per (unit of input). A population in thousands measured in hours gives thousand cells per hour. On a free-response question the units are worth a point on their own.

Second differences and degree

Constant first differences means linear; constant second differences means quadratic; constant nth differences means degree n. Requires equally spaced inputs to be meaningful.

Concave up from a table

Compute successive differences in f. If they are growing, the rate of change is increasing and the graph is concave up. Shrinking differences mean concave down.

Point of inflection on a growth curve

The moment of fastest growth: growth has been accelerating and is about to start slowing. It is where concavity changes, not where the function turns around.

Why concavity is not about direction

A decreasing function can be concave up — a decay curve flattening toward its asymptote falls the whole way while bending upward. Concavity is judged from the rate of change, never from whether the graph rises.

Factor Theorem

(x − a) is a factor of p exactly when p(a) = 0. It is the Remainder Theorem with remainder zero, and it is why finding zeros and finding factors are the same task.

Placeholder coefficients in synthetic division

Every power needs an entry, including zeros for missing terms. Dividing 2x³ − 3x² − 4 requires the row 2, −3, 0, −4. Omitting the placeholder is the standard error.

Reducing a cubic with a known zero

Divide the known factor out to get a quadratic, then factor or use the quadratic formula. Each division drops the degree by one, turning an intractable problem into a familiar one.

Odd-degree real polynomial

Must have at least one real zero. Non-real zeros come in conjugate pairs, so they are even in number and cannot account for an odd degree by themselves.

Reading multiplicity from a graph

Crossing straight through means multiplicity 1. Touching and turning back means even multiplicity. Crossing with a visible flattening means odd multiplicity of 3 or more.

Bounds on the number of real zeros

Counting x-intercepts gives a lower bound only. Degree minus (real zeros with multiplicity) is the number of non-real zeros, and that difference is always even.

Sign change and multiplicity

A polynomial changes sign only at zeros of odd multiplicity. At an even-multiplicity zero it touches zero and returns to the same side, so a sign chart that ignores multiplicity gets the answer wrong.

Test point per interval

A polynomial cannot change sign inside an interval containing no zeros, so one test value settles that whole interval. This is what makes sign charts finite work.

Why you cannot multiply out a rational inequality

The denominator's sign is unknown, and multiplying by a negative reverses the inequality. Subtract to get zero on one side and combine over a common denominator instead.

Endpoint inclusion in an inequality solution

Numerator zeros are included when the inequality allows equality. Denominator zeros are NEVER included — the expression has no value there at all.

Coordinates of a hole

Cancel the common factor, then evaluate the simplified expression at that x. Substituting into the original gives 0/0 and tells you nothing.

Why a calculator hides a hole

A grapher samples finitely many x values and connects them. Unless a sample lands exactly on the hole, the curve is drawn straight through it. Holes are found by factoring, never by graphing.

Removable vs non-removable discontinuity

A hole is removable: defining one extra point would make the function continuous. A vertical asymptote is not — no single value can be assigned there.

Partial cancellation in a rational function

If a factor appears twice in the denominator and once in the numerator, one copy survives — so it is an asymptote, not a hole. Cancel first, then look at what is actually left.

Crossing a horizontal asymptote

Perfectly legal. An asymptote describes end behavior only, so a graph may cross it any number of times in the middle. Only vertical asymptotes are never crossed, because the function is undefined there.

End-behavior model of a rational function

The ratio of leading terms. (2x⁵ − x)/(x² + 3) behaves like 2x³ for large |x|. This single idea reproduces all three asymptote cases and handles the ones with no asymptote at all.

Domain of a composition

x must be in the domain of the inner function AND the inner output must be in the domain of the outer. The second condition is the one that gets dropped.

Decomposing a function

Ask what happens LAST. For (3x + 1)⁵ the last step is the fifth power, so f(u) = u⁵ is outer and g(x) = 3x + 1 is inner.

Solving for the inner function

Given f and f∘g, treat g(x) as a single unknown and solve algebraically. If f(x) = 2x + 1 and f(g(x)) = 2x² − 5, then 2g(x) + 1 = 2x² − 5, so g(x) = x² − 3.

Leading coefficient and the right arm

Positive lead sends the right arm up; negative sends it down. The degree's parity then decides whether the left arm agrees (even) or disagrees (odd).

Why a degree-n polynomial has at most n − 1 turns

Each turning point is a place the rate of change switches sign, and the rate of change is a polynomial of degree n − 1, which has at most n − 1 zeros. Fewer turns is possible; more is not.

Polynomial from its zeros

Multiply the factors (x − rᵢ). The zeros determine the polynomial only up to a constant multiple, so any nonzero scalar multiple is equally correct.

Sum of a conjugate pair

(a + bi) + (a − bi) = 2a, real. Their product is a² + b², also real. This is why real polynomials factor into real linear and real quadratic pieces.

Real quadratic from a conjugate pair

Group as [(x − a) + bi][(x − a) − bi] = (x − a)² + b². Multiplying the conjugate pair first turns it into a difference of squares, so no i survives to be lost.

Degree from a finite difference table

Take differences until they become constant. If the third differences are constant, the data is cubic. This works only for equally spaced inputs.

Interpreting a negative average rate of change

The quantity decreased on average over the interval. It says nothing about whether it decreased throughout — the function may have risen and fallen in between.

Relative vs absolute extremum

An absolute extremum is the largest or smallest value on the whole domain; a relative extremum is largest or smallest only nearby. A relative maximum need not be the highest point anywhere.

Increasing on an interval, definition

a < b implies f(a) < f(b) for every pair in the interval. Increasing is a claim about all pairs, not about a few sample points.

Vertical line test

A graph represents a function only if no vertical line meets it more than once. A circle fails it, which is why x² + y² = 25 defines a relation and not a function.

Horizontal line test

A function is one-to-one, and therefore invertible, only if no horizontal line meets its graph more than once. This is the test that decides whether an inverse exists.

Composition is not commutative

f∘g and g∘f are generally different in both formula and domain. For f(x) = √x and g(x) = x − 3, one is √(x − 3) with domain x ≥ 3 and the other is √x − 3 with domain x ≥ 0.

Parity of a product

even·even = even, odd·odd = even, even·odd = odd — the same arithmetic as adding exponents. x³ (odd) times x² (even) is x⁵, which is odd.

Parity of a sum

even + even = even and odd + odd = odd, but even + odd is NEITHER. x³ + 1 is neither despite looking almost odd, because the constant is an even power.

The only function both even and odd

f(x) = 0, the only function equal to its own negative. Every other function is even, odd, or neither.

Constant term and parity

A constant counts as x⁰, an even power. So adding a nonzero constant to an odd function destroys its oddness, though adding one to an even function keeps it even.

Residual plot patterns

A U shape means missing curvature. A steady drift means a missing trend. A widening fan means the spread grows with x. Featureless scatter is the only pattern that supports the model.

Overfitting

A degree-n polynomial passes exactly through any n + 1 points, so a perfect fit carries no information about whether the model is right. It has fitted the noise and will behave erratically between points.

Why r² does not validate a model

A curved relationship can produce r near 0.99 while the residual plot shows an obvious arch. The summary statistic hides the very defect it is being used to test for.

Extrapolation risk

A model fitted on one range carries no guarantee outside it. Exponential growth fitted to early data overpredicts badly once a ceiling binds.

Interpolation vs extrapolation

Interpolation estimates inside the observed range and is usually defensible. Extrapolation goes outside it and requires a reason to believe the pattern continues.

Piecewise function domain pieces

The pieces must cover the intended domain without overlapping. Continuity at a boundary requires the two adjacent rules to agree in value there.

Rational function x-intercepts

Numerator zeros that are still in the domain. A numerator zero that cancels is a hole, not an intercept — the point is missing from the graph.

Vertical asymptote is never crossed

The function is undefined there, so no point of the graph has that x value. Horizontal and slant asymptotes can be crossed freely, since they describe end behavior only.

Choosing a model from a mechanism

Justify the family by the structure: constant increments give linear, constant factors give exponential, a fixed cycle gives sinusoidal. Fit quality alone never justifies a model.

What examiners penalize here

• On the AP exam, state end behavior with limit-style language: “as x → ∞, f(x) → ∞.” Naming the degree parity and the sign of the leading coefficient is the reasoning that earns the point.
• When a graph is given, read multiplicity from the shape: a straight-through crossing is multiplicity 1, a flattened crossing is multiplicity 3+, and a bounce is even. This lets you rebuild the factored form directly from a picture.
• Always factor fully first. The exam loves a rational function whose denominator “looks like” two asymptotes but where one factor cancels into a hole. Simplify, then read vertical asymptotes and holes off the reduced form.
• When several transformations combine, handle inside changes (horizontal shifts, y-axis flips) separately from outside changes (vertical shifts, x-axis flips). Describe each precisely — the AP rubric awards the direction and the axis, not just the word “shift.”
• When a free-response question says "average rate of change," it wants a computation and units. When it says "describe the rate of change," it usually wants *increasing or decreasing, and whether that rate is itself increasing or decreasing* — a different question, taken up in the next lesson.
• Free-response questions ask you to *justify* concavity, not just assert it. A justification cites evidence: "the average rates of change on successive intervals are 3, 6, 9, 12, which are increasing, so f is concave up." Naming the evidence is where the point is earned.
• Non-real zeros are examinable on AP Precalculus, but building polynomials from complex zeros shows up far more often than computing with i. Practice the reconstruction direction — given some zeros, produce the polynomial.
• A favorite exam setup: a rational function whose x-intercept appears to be at the hole. In the worked example above, x = −2 makes the numerator zero, yet it is **not** an x-intercept — the point is missing from the graph. Always check candidate intercepts against the domain.
• Symmetry halves your work. If you know a function is even, analyzing x ≥ 0 determines the whole graph by reflection. Free-response questions sometimes hand you symmetry precisely so you can transfer a computed value from one side to the other.
• When a question asks you to "justify the choice of model," a numerical fit is not enough for full credit. Name the structural feature — constant ratio, constant second difference, fixed period — and, where the context allows, the reason that feature makes sense.

Practice Precalculus

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 4 units of AP Precalculus

1. Unit 1 · Polynomial & Rational Functions
2. Unit 2 · Exponential & Logarithmic Functions
3. Unit 3 · Trigonometric & Polar Functions
4. Unit 4 · Functions Involving Parameters, Vectors & Matrices