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

Formulas in Unit 1

Every term in Unit 1

All 37 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.

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.”

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