SAT / PSAT
SAT / PSAT Prep
History & Social Science
AP World History AP US History AP European History AP Human Geography AP US Government & Politics AP Psychology AP Macroeconomics AP Microeconomics
English
AP English Language & Composition AP English Literature & Composition
Math & Computer Science
AP Calculus AB/BC AP Precalculus AP Statistics AP Computer Science A AP Computer Science Principles
Sciences
AP Biology AP Chemistry AP Environmental Science AP Physics 1 AP Physics 2
World Languages & Arts
AP Spanish Language AP Art History AP Music Theory Start studying →
Unit 1 · Polynomial & Rational Functions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Precalculus Unit 1 Visual Review

A topic-by-topic visual walkthrough of Unit 1: Polynomial and Rational Functions — rates of change, polynomial end behavior and zeros, rational functions, transformations, and function modeling.

← Back to Unit 1 hub
TOPIC 1.1 Change in Tandem Two quantities change together A function pairs each input with exactly one output: as x changes, y changes. INCREASING: y goes up as x goes up. CONCAVE UP: rate of change is increasing. Example V(t) = filling a cone with water As height h rises at a steady rate, the volume V rises FASTER and faster. → V increasing & concave up in h. Describe change from a graph, table, or context Track WHERE a function increases/decreases and its concavity. A point where concavity switches is an INFLECTION POINT. Increasing + concave down still rises, but by smaller and smaller amounts. Track how output changes with input — increasing/decreasing and concavity. The Review Hub · AP Precalculus Unit 1 TOPIC 1.2 Rates of Change avg rate of change = Δy / Δx = [f(b) − f(a)] / (b − a) Worked example f(x) = x², from a = 1 to b = 3: = (f(3) − f(1)) / (3 − 1) = (9 − 1) / 2 = 4 This is the SLOPE of the secant line through (1, 1) and (3, 9). Average vs. instantaneous AVERAGE rate = slope of the SECANT line between two points. INSTANTANEOUS rate = slope of the TANGENT line at a single point. Estimate the instantaneous rate with a small Δx. Average rate of change = Δy/Δx = slope of the secant line. The Review Hub · AP Precalculus Unit 1 TOPIC 1.3 Rates of Change: Linear & Quadratic Linear: constant rate f(x) = mx + b The rate of change m is CONSTANT. Equal x-steps give equal y-steps. y = 2x+1: +1 in x → +2 in y always Second differences = 0. Quadratic: rate changes linearly f(x) = ax² + bx + c The rate of change is NOT constant, but the SECOND differences are constant. y = x²: 1st diffs 1,3,5,7… 2nd diffs 2,2,2… (constant) Use differences to identify the model Over equal x-intervals: constant 1st differences → LINEAR; constant 2nd differences → QUADRATIC. Linear → constant 1st differences; quadratic → constant 2nd differences. The Review Hub · AP Precalculus Unit 1 TOPIC 1.4 Polynomial Functions & Rates of Change Local behavior p(x) = aₙxⁿ + … + a₁x + a₀ A LOCAL MAX/MIN (extremum) is where the function switches increasing ↔ decreasing. An inflection point is where concavity switches. Degree & turning points A degree-n polynomial has AT MOST n − 1 turning points and n real zeros. p(x)=x³−3x → up to 2 turns The rate of change of a polynomial is itself a polynomial of degree n − 1. Rate of change is greatest at inflection points Between two consecutive extrema, a polynomial has exactly one inflection point where its rate of change is extreme. 〽Degree n → ≤ n−1 turning points; extrema mark increasing↔decreasing switches. The Review Hub · AP Precalculus Unit 1 TOPIC 1.5 Polynomial Functions & Complex Zeros degree n ⟹ exactly n complex zeros (with multiplicity) Multiplicity & the graph p(x) = (x−2)²(x+1) EVEN multiplicity (x−2)² → graph TOUCHES the x-axis at x = 2. ODD multiplicity (x+1) → graph CROSSES the x-axis at x = −1. Complex zeros come in pairs Non-real zeros of a real polynomial occur in CONJUGATE pairs a ± bi. x² + 4 = 0 → x = ±2i So a real cubic has 1 or 3 real zeros — never exactly 2. Even multiplicity touches, odd crosses; complex zeros come in conjugate pairs. The Review Hub · AP Precalculus Unit 1 TOPIC 1.6 Polynomial Functions & End Behavior x→−∞ x→+∞ odd degree, positive leading coeff. Leading term controls the ends End behavior is set by aₙxⁿ (highest term). EVEN degree: both ends go the SAME way. ODD degree: ends go OPPOSITE ways. Limit notation lim(x→∞) x³ = +∞ lim(x→−∞) x³ = −∞ Negative leading coeff flips both ends. ↗The leading term sets end behavior: even = same ends, odd = opposite ends. The Review Hub · AP Precalculus Unit 1 TOPIC 1.7 Rational Functions & End Behavior r(x) = p(x) / q(x) — compare degrees for the end behavior deg p < deg q Horizontal asymptote y = 0 1/(x+2) bottom-heavy → 0 deg p = deg q HA = ratio of leading coefficients y = aₙ / bₙ (2x²+1)/(x²) → y=2 deg p > deg q No horizontal asymptote. If exactly one degree higher → SLANT asymptote. (x²+1)/x → y = x Find it by division. Compare degrees: < → y=0, = → ratio of leading coeffs, > → slant/none. The Review Hub · AP Precalculus Unit 1 TOPIC 1.8 Rational Functions & Zeros Zeros come from the NUMERATOR r(x) = 0 exactly where the numerator p(x) = 0 (and q(x) ≠ 0 there). r(x) = (x−3)/(x+1) zero at x = 3 (crosses the x-axis). A zero is the x-intercept of the graph. Sign analysis Between consecutive zeros and vertical asymptotes, r(x) keeps a constant SIGN. Test a point in each interval to find where the graph is above/below the axis. Even multiplicity zero → touches, doesn't cross. Watch for common factors If a factor cancels with the denominator, that x-value is a HOLE (removable), not a zero — see Topic 1.10. Rational zeros occur where the numerator = 0 (and the denominator ≠ 0). The Review Hub · AP Precalculus Unit 1 TOPIC 1.9 Rational Functions & Vertical Asymptotes x = a graph shoots to ±∞ at the asymptote From the DENOMINATOR A vertical asymptote occurs where q(x) = 0 but p(x) ≠ 0 (no cancellation). r(x)=1/(x−2) → VA at x = 2 Behavior near the asymptote As x → a from each side, r(x) → +∞ or −∞. lim(x→2⁺) 1/(x−2) = +∞ lim(x→2⁻) 1/(x−2) = −∞ Vertical asymptotes occur where the denominator = 0 and doesn't cancel. The Review Hub · AP Precalculus Unit 1 TOPIC 1.10 Rational Functions & Holes r(x) = (x−3)(x+1) / (x−3) = x + 1, x ≠ 3 A hole = a canceled factor When the SAME factor is in the top and bottom, it cancels → a removable hole. Here (x−3) cancels, so there's a HOLE at x = 3, not a vertical asymptote. The domain still excludes x = 3. Find the hole's coordinates 1. Cancel the common factor. 2. Plug the excluded x into what's left. at x=3: y = 3+1 = 4 Hole at the point (3, 4) — an open circle on the graph. A canceled common factor makes a hole (removable) — not an asymptote. The Review Hub · AP Precalculus Unit 1 TOPIC 1.11 Equivalent Representations of Expressions Factored vs. standard form x² − x − 6 = (x−3)(x+2) FACTORED form reveals the zeros (x = 3 and x = −2) at a glance. STANDARD form reveals the y-intercept (the constant term, here −6). Polynomial division & remainder Rewrite a rational expression as quotient + remainder/divisor REMAINDER THEOREM: p(a) equals the remainder when p(x) is divided by (x−a). If p(a)=0 then (x−a) is a factor. Equivalent forms expose different features Choose the form that shows what you need — factored for zeros, standard for intercepts, vertex form for the vertex. Factored shows zeros, standard shows intercepts; remainder theorem: p(a) = remainder. The Review Hub · AP Precalculus Unit 1 TOPIC 1.12 Transformations of Functions g(x) = a · f( b(x − h) ) + k Outside affects y (vertical) a: vertical stretch (|a|>1) / shrink; a < 0 reflects over the x-axis. k: vertical shift up (+) / down (−). 2f(x)+3 → stretch ×2, up 3 Vertical changes act as written. Inside affects x (horizontal) h: horizontal shift RIGHT (+h). b: horizontal shrink by 1/b; b < 0 reflects over the y-axis. Horizontal changes are OPPOSITE/inverse. f(x−4) → shifts RIGHT 4 Outside = vertical (as written); inside = horizontal (opposite). The Review Hub · AP Precalculus Unit 1 TOPIC 1.13 Function Model Selection Match the model to the pattern of change LINEAR: constant rate. QUADRATIC: constant 2nd differences. POLYNOMIAL: several turning points. Choose the simplest function whose behavior fits the data and the real-world context. State your assumptions A model is only valid under stated assumptions and over a DOMAIN that makes sense for the context. e.g. time t ≥ 0; a population can't be negative. Note any restrictions on the outputs too. Judge the fit A good model captures the overall trend and concavity, not just single points. RESIDUALS (actual − predicted) should be small with no leftover pattern. A pattern in residuals means a poor model choice. Pick the model that fits the pattern of change; state assumptions and a valid domain. The Review Hub · AP Precalculus Unit 1 TOPIC 1.14 Function Model Construction & Application Build a model from information Construct a function from a graph, table, verbal description, or data set. Use known features — zeros, intercepts, asymptotes, a vertex — to find the parameters. Apply it: predict & solve Once built, use the model to answer questions about the context. Given zeros −2, 1, 3 & point: f(x)=a(x+2)(x−1)(x−3) Solve for a using the extra point. Interpret answers in context Always give units and check the answer is reasonable for the situation (within the valid domain and range). Build from known features, then apply the model to predict and interpret in context. The Review Hub · AP Precalculus Unit 1
1 / 14

How to use the visual review

Spend 30 seconds per slide before clicking next. Look at the graph, then ask yourself: "Could I sketch this from memory and explain what its algebra is doing?"

Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.

This is great for review the night before the exam — fast, visual, and covers every graph feature you need to recognize in Unit 1.