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Unit 1 · Polynomial & Rational Functions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Precalculus Unit 1 Essentials

The must-know terms and core concepts for Unit 1: Polynomial & Rational Functions. Every vocabulary word and idea you need to master.

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Key Concept 1
A function describes two quantities changing in tandem — and we describe that change precisely
A function maps each input to exactly one output, and the whole course is about how those outputs change as the inputs change. Unit 1 gives you the vocabulary to describe change carefully: where a function increases or decreases, whether it is concave up or concave down, its average rate of change over an interval, and its rate of change at a single point. Precise language — "increasing at a decreasing rate," not "goes up" — is exactly what earns points on the exam.
Change in Tandem Rates of Change Concavity
Key Concept 2
A function's algebraic structure determines its graphical behavior
Almost every graph feature traces back to the algebra. The factors of a polynomial give its zeros, and each zero's multiplicity decides whether the graph crosses or just touches the x-axis. The leading term alone sets the end behavior. For rational functions, the denominator produces vertical asymptotes (where it is zero and the numerator is not) and holes (where a factor cancels), while comparing the degrees of numerator and denominator gives the horizontal or slant asymptote. Learn the "why" behind each feature and you never have to memorize graphs.
Zeros & Multiplicity End Behavior Asymptotes & Holes
Key Concept 3
Functions live in four representations — and modeling means owning your assumptions
Every function can be expressed graphically, numerically, analytically, and verbally, and the exam constantly asks you to move between them and to rewrite expressions into equivalent forms that reveal features. Modeling adds one more skill: choosing an appropriate function type for a data set (constant rate → linear, changing rate → polynomial, asymptotic behavior → rational), then stating the assumptions the model depends on, such as a restricted domain or the range over which it is valid.
Multiple Representations Transformations Modeling
Function
A mathematical relation that maps each input value to exactly one output value. The set of inputs is the domain; the set of outputs is the range.
Change in Tandem
Increasing / decreasing
A function is increasing on an interval if larger inputs always give larger outputs (a < b ⟹ f(a) < f(b)), and decreasing if larger inputs always give smaller outputs.
Change in Tandem
Concave up / concave down
A graph is concave up on intervals where the rate of change is increasing, and concave down where the rate of change is decreasing.
Change in Tandem
Zero of a function
An input value where the output is zero. Real zeros appear as the x-intercepts of the graph.
Change in Tandem
Average rate of change
Over an interval [a, b], it equals (f(b) − f(a))/(b − a) — the slope of the secant line joining the endpoints.
Rates of Change
Rate of change at a point
The instantaneous rate of change at a single input, estimated by the slope of the tangent line or by average rates over very small intervals.
Rates of Change
Point of inflection
A point where the graph changes concavity — the rate of change switches from increasing to decreasing, or vice versa.
Rates of Change
Local maximum / minimum
A point where a function changes from increasing to decreasing (local max) or from decreasing to increasing (local min). Also called a relative extremum.
Rates of Change
Polynomial function
A function that is a sum of terms of the form (coefficient)·x^n with whole-number exponents. Its degree is the highest exponent.
Polynomials
Complex zero
A zero of the form a + bi. A degree-n polynomial has exactly n complex zeros counting multiplicity (Fundamental Theorem of Algebra). Non-real zeros occur in conjugate pairs.
Polynomials
Multiplicity
The number of times a factor (x − a) appears in a polynomial. At odd multiplicity the graph crosses the x-axis; at even multiplicity it touches and turns around.
Polynomials
Leading term
The term of highest degree. Its degree and coefficient sign alone determine the polynomial's end behavior.
Polynomials
End behavior
What the output values approach as the input increases or decreases without bound (x → ±∞). Expressed with limit notation such as lim(x→∞) f(x) = ∞.
Polynomials
Rational function
A quotient of two polynomials, p(x)/q(x), with q(x) not the zero polynomial. Its behavior comes from comparing the numerator and denominator.
Rational Functions
Vertical asymptote
A vertical line x = a that the graph approaches without bound. It occurs where the denominator is zero and the numerator is not (after canceling common factors).
Rational Functions
Horizontal asymptote
The end-behavior line of a rational function. y = 0 if the numerator's degree is smaller; y = ratio of leading coefficients if the degrees are equal.
Rational Functions
Slant (oblique) asymptote
A diagonal end-behavior line that occurs when the numerator's degree is exactly one more than the denominator's. Found by polynomial division.
Rational Functions
Hole (removable discontinuity)
A single missing point in a graph caused by a factor common to numerator and denominator that cancels.
Rational Functions
Equivalent representations
Different analytic forms of the same expression — factored, standard, or quotient form — each chosen to reveal specific features like zeros or asymptotes.
Representations
Transformation
A change to a parent function's rule that shifts, stretches, compresses, or reflects its graph.
Transformations
Additive transformation (translation)
Adding a constant translates the graph: f(x) + k shifts up/down by k; f(x − h) shifts right by h. Inside changes act opposite to their sign.
Transformations
Multiplicative transformation (dilation)
Multiplying scales the graph: a·f(x) is a vertical dilation by |a|; f(bx) is a horizontal dilation by 1/|b|. A negative factor also reflects the graph.
Transformations
Function model
A function chosen to represent a real-world data set, together with the assumptions it relies on (such as a restricted domain or a limited range of validity).
Modeling
Regression
A technology-based procedure that fits a function of a chosen type to a data set; residuals are analyzed to judge how appropriate the model is.
Modeling