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Every rule and concept in Precalculus, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (186)

$\cos 36^\circ$ from a cubic equation: Not every “nice” angle needs a sum/difference split — some fall out of a polynomial.; Analytic Trigonometry › Going Deeper: Advanced Analytic Trig
A bigger toolbox for the $\tfrac{0}{0}$ form: When direct substitution gives $\tfrac{0}{0}$ , the expression hides a common factor you must expose and cancel.; Introduction to Limits › Going Deeper: Advanced Limit Ideas
A function is a machine: A function takes an input, does something to it, and gives one output.; Core Ideas in Plain Terms › Functions and Transformations
A limit is where a value is heading: A limit asks: as the input gets closer and closer to some number, what does the output approach?; Core Ideas in Plain Terms › Sequences, Series, and Limits
Adding, scaling, multiplying: For matrices of the same size, add/subtract entrywise; scalar multiply multiplies every entry.; Systems, Matrices, and Determinants › Matrices and Their Operations
Anatomy of an ellipse: Standard form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b$ ): the long semi-axis is $a$ , the short one $b$ , and the two foci sit at $(\pm c, 0)$ with $c^2 = a^2 - b^2$ .; Conic Sections and Parametric Equations › The Ellipse
Angles, radians & arc length: $180^\circ=\pi$ rad.; Course Review › Trigonometric Functions
Approach: To evaluate something like $\sin(\arccos u)$ , let $\theta=\arccos u$ , so $\cos\theta=u$ with $\theta$ in the range of arccosine ( $[0,\pi]$ ).; Analytic Trigonometry › Combining Identities with Inverse Trig Functions
Arctangent sums and Machin-style identities: The addition law, read backward, adds arctangents: \[ a+ b= $\frac{a+b}{1-ab}$ (mod π), \] valid outright when $ab<1$ .; Analytic Trigonometry › Going Deeper: Advanced Analytic Trig
Area by several methods, and area ratios: The same triangle's area can be written many ways; equating them yields identities.; Triangle Trigonometry and Vectors › Going Deeper: Advanced Triangle Trig & Vectors
Area enclosed by a polar curve: A thin wedge of angle $d\theta$ at radius $r$ is a circular sector of area $\tfrac12 r^2\,d\theta$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Arithmetic & geometric: Arithmetic (common difference $d$ ): $a_n=a_1+(n-1)d$ ; sum $S_n=\dfrac n2(a_1+a_n)=\dfrac n2\big[2a_1+(n-1)d\big]$ .; Course Review › Sequences, Series & the Binomial Theorem
Arithmetic and composition: For functions $f$ and $g$ : \[ (f± g)(x)=f(x)± g(x), (fg)(x)=f(x) g(x), ( $\tfrac{f}{g}$ )(x)= $\frac{f(x)}{g(x)}$ , g(x)≠ 0.; Functions and Their Graphs › Combinations and Composition of Functions
Arithmetic sequences: An arithmetic sequence adds a fixed common difference $d$ each step, so $d=a_{n}-a_{n-1}$ .; Sequences, Series, and the Binomial Theorem › Arithmetic Sequences and Partial Sums
Asymptote rules: For $f(x)=\dfrac{N(x)}{D(x)}$ in lowest terms: Domain: all reals except the zeros of $D(x)$ .; Polynomial and Rational Functions › Rational Functions: Domain, Asymptotes, Holes, and Sketching
Asymptotes and Periods: Trigonometric Functions › Graphs of Tangent, Cotangent, Secant, Cosecant
Asymptotes are the skeleton: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ opens left-right with foci at $(\pm c, 0)$ , $c^2 = a^2 + b^2$ (note the PLUS — foci sit outside), and asymptotes $y = \pm\tfrac{b}{a}x$ : the branches hug these lines forever without touching.; Conic Sections and Parametric Equations › The Hyperbola
Average rate of change: The average rate of change of $f$ from $x=a$ to $x=b$ is the slope of the secant line: \[ $\frac{f(b)-f(a)}{b-a}$ .; Functions and Their Graphs › Piecewise Functions and Average Rate of Change
Basic probability: For equally likely outcomes, \[ P(E)=number of favorable outcomestotal number of outcomes= $\frac{n(E)}{n(S)}$ .; Sequences, Series, and the Binomial Theorem › Probability Using Counting
Binomial coefficients and the theorem: The binomial coefficient is \[ $\binom{n}{k}$ = $\frac{n!}{k!\,(n-k)!}$ , $\binom{n}{0}$ = $\binom{n}{n}$ =1 .; Sequences, Series, and the Binomial Theorem › The Binomial Theorem & Pascal's Triangle
Binomial Theorem: \[ (a+b)^n=∑_k=0^n $\binom{n}{k}$ a^ n-kb^k.; Course Review › Sequences, Series & the Binomial Theorem
Binomial-coefficient identities: These follow from the Binomial Theorem and from Pascal's rule $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ :; Sequences, Series, and the Binomial Theorem › Going Deeper: Advanced Sequences, Series & Binomial
Breaking a rational function apart: For a proper fraction (numerator degree $<$ denominator degree), factor the denominator and write one term per factor:; Systems, Matrices, and Determinants › Partial Fraction Decomposition
Building a Sinusoid from Data: Given a periodic phenomenon with maximum value $M$ , minimum value $m$ , and known timing, model it as $y=a\cos\!\big(b(t-c)\big)+d$ (or with $\sin$ ): \[ d= $\frac{M+m}{2}$ (midline), a= $\frac{M-m}{2}$ (amplitude), b=2πp…; Trigonometric Functions › Going Deeper: Advanced Trig Ideas
Change-of-base chains and telescoping products: Change of base can be strung into a chain.; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Circular Arcs and Sectors: For a circle of radius $r$ and a central angle $\theta$ measured in radians: \[ Arc length: s=rθ Sector area: A= $\tfrac{1}{2}$ r^2θ.; Trigonometric Functions › Angles: Degree and Radian Measure
Classifying a general conic by $B^2-4AC$: The full second-degree equation \[ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \] may include a cross term $Bxy$ , which rotates the axes.; Conic Sections and Parametric Equations › Going Deeper: Advanced Conics & Parametrics
Classifying discontinuities: Introduction to Limits › Continuity at a Point; Types of Discontinuity
Complex numbers & trig form: $z=a+bi$ , $|z|=r=\sqrt{a^2+b^2}$ .; Course Review › Polar Coordinates & Complex Numbers
Complex zeros come in pairs: With $i^2=-1$ , a complex number is $a+bi$ and its conjugate is $a-bi$ ; note $(a+bi)(a-bi)=a^2+b^2$ .; Polynomial and Rational Functions › Complex Numbers and the Fundamental Theorem of Algebra
Constructing polynomials from constraints; roots in AP or GP: From constraints: a degree- $n$ polynomial has $n+1$ coefficients, so $n+1$ independent conditions (values, zeros, matching derivatives/slopes, a leading coefficient) determine it.; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
Continuity & the derivative: $f$ is continuous at $c$ if all hold: (1) $f(c)$ defined, (2) $\displaystyle\lim_{x\to c}f(x)$ exists, (3) $\displaystyle\lim_{x\to c}f(x)=f(c)$ .; Course Review › Introduction to Limits
Continuous growth versus logistic (bounded) growth: Pure continuous growth $A=A_0e^{kt}$ is unbounded, so it only models the early phase of real populations.; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Coterminal, Complementary, Supplementary: Trigonometric Functions › Angles: Degree and Radian Measure
Counting Solutions in $[0,2\pi)$: To count solutions of a trig equation on one period, reduce it to $F(u)=k$ where $u=n\theta$ :; Trigonometric Functions › Going Deeper: Advanced Trig Ideas
Curves defined by a parameter: A parametric curve gives $x$ and $y$ each as a function of a third variable $t$ : \[ x=f(t), y=g(t).; Conic Sections and Parametric Equations › Parametric Equations
Cutting both nappes: a hyperbola: Tilt past the parallel position (or cut straight down) and the plane slices BOTH halves of the double cone, producing the two branches of a hyperbola.; Conic Sections and Parametric Equations › Where Conics Come From: Slicing a Cone
Definition and key features: An exponential function has the form \[ f(x)=a b^x, a≠ 0, b>0, b≠ 1 .; Exponential and Logarithmic Functions › Exponential Functions and Their Graphs
Definition on the Unit Circle: On the unit circle ( $x^2+y^2=1$ ), the terminal side of angle $\theta$ meets the circle at $(\cos\theta,\sin\theta)$ .; Trigonometric Functions › The Unit Circle and the Six Trig Functions
Definition, common and natural logs: For $b>0,\ b\neq 1$ , the logarithm is the inverse of the exponential: \[ y=_b x b^y=x .; Exponential and Logarithmic Functions › Logarithmic Functions and Their Graphs
Definition, domain & range: A function $f$ assigns each input $x$ (the domain) exactly one output $y=f(x)$ ; the outputs form the range.; Course Review › Functions and Their Graphs
Definitions, Domains, and Ranges: To be invertible, each trig function is restricted: \[; Trigonometric Functions › Inverse Trigonometric Functions
Determinants: $2\times2$ : $; Systems, Matrices, and Determinants › Determinants and Cramer's Rule
Difference quotient: The difference quotient of $f$ is \[ $\frac{f(x+h)-f(x)}{h}$ , h≠ 0.; Functions and Their Graphs › The Difference Quotient
Difference quotient, rate of change, composition, inverses: Difference quotient: $\dfrac{f(x+h)-f(x)}{h},\ h\ne 0$ .; Course Review › Functions and Their Graphs
Division algorithm and two theorems: For polynomials, $f(x)=d(x)\,q(x)+r(x)$ with $\deg r<\deg d$ .; Polynomial and Rational Functions › Polynomial Long Division; Remainder & Factor Theorems
Division theorems & rational zeros: Division algorithm: $f(x)=d(x)q(x)+r(x)$ .; Course Review › Polynomial and Rational Functions
Dot product and its uses: For $\vec u=\langle a,b\rangle$ and $\vec v=\langle c,d\rangle$ : \[ u· v=ac+bd=\| u\| \| v\|θ.; Triangle Trigonometry and Vectors › The Dot Product, Angle, Projection, and Work
Eccentricity & parametric equations: Eccentricity $e=\dfrac ca$ : circle $e=0$ ; ellipse $0<e<1$ ; parabola $e=1$ ; hyperbola $e>1$ .; Course Review › Conic Sections & Parametric Equations
Eccentricity form: With a focus at the pole, a conic has polar equation \[ r= $\frac{ed}{1\pm e\cos\theta}$ r= $\frac{ed}{1\pm e\sin\theta}$ , \] where $e$ is the eccentricity and $d>0$ is the distance from focus to directrix.; Conic Sections and Parametric Equations › Polar Equations of Conics (Introduction)
Eight parents to memorize: Constant $f(x)=c$ ; Identity $f(x)=x$ ; Absolute value $f(x)=|x|$ ; Square root $f(x)=\sqrt{x}$ ; Quadratic $f(x)=x^2$ ; Cubic $f(x)=x^3$ ; Reciprocal $f(x)=\dfrac{1}{x}$ ; Greatest integer (step) $f(x)=\lfloor x\rfloor$ .; Functions and Their Graphs › The Library of Parent Functions
End behavior & zeros: For leading term $a_nx^n$ : if $n$ even, both ends same way ( $a_n>0$ up/up, $a_n<0$ down/down); if $n$ odd, ends opposite ( $a_n>0$ down/up).; Course Review › Polynomial and Rational Functions
Exact Values at Special Angles: \[; Trigonometric Functions › The Unit Circle and the Six Trig Functions
Exponential equations that hide a quadratic: When an equation contains a base raised to $x$ and to $2x$ (or $-x$ ), substitute $u=b^{x}$ (with $u>0$ ) to expose a quadratic: \[ b^2x=(b^x)^2=u^2, b^-x=1u .; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Exponential functions and $e$: $f(x)=a\,b^{x}$ ( $b>0,\ b\ne1$ ): horizontal asymptote $y=0$ , domain $(-\infty,\infty)$ , range $(0,\infty)$ ; grows if $b>1$ , decays if $0<b<1$ .; Course Review › Exponential and Logarithmic Functions
Expressing logs in terms of given letters: If you are told, e.g., $\log_2 3=a$ and $\log_2 5=b$ , then every log whose argument factors over $\{2,3,5\}$ can be written in $a,b$ using the log laws — and change of base for a different outer base.; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Feasible regions and optimization: Graph each inequality and shade the overlap—the feasible region.; Systems, Matrices, and Determinants › Systems of Inequalities and Linear Programming
Focal chords and the latus rectum: A focal chord is any chord through a focus.; Conic Sections and Parametric Equations › Going Deeper: Advanced Conics & Parametrics
Focus and directrix: $x^2 = 4py$ : focus $(0, p)$ , directrix $y = -p$ , and every point of the curve is EQUIDISTANT from the focus and the directrix.; Conic Sections and Parametric Equations › The Parabola
Formulas: Analytic Trigonometry › Sum and Difference Formulas
From secant slopes to the tangent slope: The slope of the secant line through $\big(x,f(x)\big)$ and $\big(x+h,f(x+h)\big)$ is the difference quotient $\dfrac{f(x+h)-f(x)}{h}$ .; Introduction to Limits › The Tangent-Line Problem, the Derivative, and the Area Problem
Functional equations: A functional equation constrains an unknown function $f$ by an identity that must hold for all allowed $x$ .; Functions and Their Graphs › Going Deeper: Advanced Function Ideas
Fundamental identities: Reciprocal/Quotient: $\csc=\tfrac1{\sin}$ , $\sec=\tfrac1{\cos}$ , $\cot=\tfrac1{\tan}=\tfrac{\cos}{\sin}$ , $\tan=\tfrac{\sin}{\cos}$ .; Course Review › Analytic Trigonometry
Fundamental Identities: Reciprocal: $\csc\theta=\dfrac{1}{\sin\theta}$ , $\sec\theta=\dfrac{1}{\cos\theta}$ , $\cot\theta=\dfrac{1}{\tan\theta}$ .; Trigonometric Functions › Right-Triangle Trig and Fundamental Identities
Generating-function-flavored counting: A product of binomials is a bookkeeping device: the coefficient of a power records how many ways to reach it.; Sequences, Series, and the Binomial Theorem › Going Deeper: Advanced Sequences, Series & Binomial
Geometric sequences and series: A geometric sequence multiplies by a fixed common ratio $r$ each step, so $r=\dfrac{a_n}{a_{n-1}}$ .; Sequences, Series, and the Binomial Theorem › Geometric Sequences & Series (incl. Infinite)
Graphs & inverse trig: For $y=a\sin(bx-c)+d$ (same for $\cos$ ): amplitude $|a|$ , period $\dfrac{2\pi}{b}$ , phase shift $\dfrac{c}{b}$ , midline $y=d$ .; Course Review › Trigonometric Functions
How an infinite series can add to a finite number: In a geometric series each term is the same fraction $r$ of the one before.; Core Ideas in Plain Terms › Going Deeper: Log Rules and Infinite Series
Indeterminate forms, infinite limits, limits at infinity: $\tfrac00$ form: factor and cancel, rationalize, or simplify, then substitute.; Course Review › Introduction to Limits
Infinite limits and vertical asymptotes: If $f(x)$ increases or decreases without bound as $x\to a$ , we write $\displaystyle\lim_{x\to a}f(x)=\infty$ (or $-\infty$ ).; Introduction to Limits › One-Sided Limits; When a Limit Fails; Infinite Limits
Inverse Composition and Range Traps: $f^{-1}(f(x))$ returns $x$ only when $x$ already lies in the restricted range of the inverse.; Trigonometric Functions › Going Deeper: Advanced Trig Ideas
Inverses: $A^{-1}$ satisfies $AA^{-1}=A^{-1}A=I$ .; Systems, Matrices, and Determinants › The Inverse of a Matrix
Iterated compositions and involutions: Composing $f$ with itself $n$ times is written \[ f^[n]=f° f°…° f_n copies, f^[1]=f, f^[n+1]=f° f^[n].; Functions and Their Graphs › Going Deeper: Advanced Function Ideas
Law of Cosines: For any triangle $ABC$ , \[ a^2=b^2+c^2-2bc A, b^2=a^2+c^2-2ac B, c^2=a^2+b^2-2ab C.; Triangle Trigonometry and Vectors › The Law of Cosines
Law of Sines: For any triangle $ABC$ , \[ $\frac{\sin A}{a}$ = $\frac{\sin B}{b}$ = $\frac{\sin C}{c}$ .; Triangle Trigonometry and Vectors › The Law of Sines
Laws of Sines and Cosines, area: Law of Sines: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ (cases AAS, ASA, SSA).; Course Review › Triangle Trigonometry & Vectors
Leading Coefficient Test and multiplicity: For $f(x)=a_nx^n+\cdots+a_0$ , the ends are controlled by the degree $n$ and sign of $a_n$ : \[; Polynomial and Rational Functions › Polynomial Functions: End Behavior, Zeros, and Sketching
Limits of the floor function $\lfloor x\rfloor$: The floor $\lfloor x\rfloor$ is the greatest integer $\le x$ ; its graph is a staircase of flat steps with jumps at every integer.; Introduction to Limits › Going Deeper: Advanced Limit Ideas
Linear programming: unbounded regions and an edge of optima: The corner principle assumes a bounded feasible region.; Systems, Matrices, and Determinants › Going Deeper: Advanced Systems & Matrices
Linked Gears and Angular-Speed Chains: Where two gears (or pulleys/wheels) mesh, the contact point shares one linear speed: $v=r_1\omega_1=r_2\omega_2$ .; Trigonometric Functions › Going Deeper: Advanced Trig Ideas
Loci in the complex plane: the Apollonius circle: Treat $z=x+iy$ as the point $(x,y)$ and read $|z-a|$ as the distance from $z$ to the point $a$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Log laws (all bases $b>0,\ b\neq1$ ; $M,N>0$ ): \[ _b(MN)=_b M+_b N _b ( $\tfrac{M}{N}$ )=_b M-_b N \] \[ _b(M^p)=p_b M _b 1=0 _b b=1 \] Change of base: $ _b M= $\frac{\log M}{\log b}$ = $\frac{\ln M}{\ln b}$ .$; Exponential and Logarithmic Functions › Properties of Logarithms and Change of Base
Log rules are exponent rules in reverse: Multiplying powers adds exponents, so logs turn multiplication into addition — the product rule, $\log(ab) = \log a + \log b$ — and pull exponents down front — the power rule, $\log(a^n) = n\log a$ .; Core Ideas in Plain Terms › Going Deeper: Log Rules and Infinite Series
Logarithmic and exponential inequalities: Solve the corresponding equation to find boundary points, then respect monotonicity and the domain:; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Logarithms & properties: $y=\log_b x \iff b^{y}=x$ (so $\log$ is the inverse of $b^{x}$ ); $\ln x=\log_e x$ .; Course Review › Exponential and Logarithmic Functions
Logarithms undo exponents: The equation $\log_b x = y$ is just another way of writing $b^y = x$ — it asks "what power turns $b$ into $x$ ?" So logs are the reverse of exponentials, and that is why they turn multiplication into addition ( $\log(xy) = \log x + \log y$ ) and pull an expone…; Core Ideas in Plain Terms › Exponentials and Logarithms
Matrix operations, inverse, determinants, Cramer: Add/subtract entrywise (same size).; Course Review › Systems, Matrices & Determinants
Matrix powers, recurrences, and the Cayley–Hamilton shortcut: Every square matrix satisfies its own characteristic equation.; Systems, Matrices, and Determinants › Going Deeper: Advanced Systems & Matrices
Meaning & limit laws: $\displaystyle\lim_{x\to c}f(x)=L$ means $f(x)$ approaches $L$ as $x\to c$ .; Course Review › Introduction to Limits
Measuring Angles: An angle in standard position has its vertex at the origin and initial side on the positive $x$ -axis.; Trigonometric Functions › Angles: Degree and Radian Measure
Method: Analytic Trigonometry › Solving Trigonometric Equations
Modulus and argument: For $z=a+bi$ : \[ r=|z|= $\sqrt{a^2+b^2}$ θ= $\frac{b}{a}$ (choose θ in the quadrant of (a,b)).; Polar Coordinates and Complex Numbers › Modulus, Argument, and Trigonometric (Polar) Form
Monotonicity, symmetry, and composed transformations: Three high-leverage ideas:; Functions and Their Graphs › Going Deeper: Advanced Function Ideas
Multiply, divide, and raise to powers: Let $z_1=r_1\,\text{cis}\,\theta_1$ and $z_2=r_2\,\text{cis}\,\theta_2$ .; Polar Coordinates and Complex Numbers › Products, Quotients, and DeMoivre's Theorem
Nonlinear systems: intersecting conics: When both equations are conics, eliminate the squared terms rather than a single variable.; Systems, Matrices, and Determinants › Going Deeper: Advanced Systems & Matrices
One cone, four curves: Take a DOUBLE cone (two ice-cream cones tip to tip, extending forever) and cut it with a plane.; Conic Sections and Parametric Equations › Where Conics Come From: Slicing a Cone
One equation, with matrices for numbers: Every linear system can be written $A\mathbf{x} = \mathbf{b}$ , where $A$ holds the coefficients, $\mathbf{x}$ the unknowns and $\mathbf{b}$ the constants.; Systems, Matrices, and Determinants › The Matrix Form $A\mathbf{x} = \mathbf{b}$
One-sided limits and existence: The right-hand limit $\displaystyle\lim_{x\to a^{+}}f(x)$ uses inputs slightly larger than $a$ ; the left-hand limit $\displaystyle\lim_{x\to a^{-}}f(x)$ uses inputs slightly smaller.; Introduction to Limits › One-Sided Limits; When a Limit Fails; Infinite Limits
One-to-one and inverses: $f$ is one-to-one if each output comes from exactly one input; equivalently it passes the horizontal line test.; Functions and Their Graphs › Inverse Functions
Parallelogram law: $\|\vec u\pm\vec v\|$ from magnitudes and the angle: If $\theta$ is the angle between $\vec u$ and $\vec v$ , then because $\|\vec u\pm\vec v\|^2=(\vec u\pm\vec v)\cdot(\vec u\pm\vec v)$ , \[ \| u+ v\|^2=\| u\|^2+\| v\|^2+2\| u\|\| v\|θ, \| u- v\|^2=\| u\|^2+\|…; Triangle Trigonometry and Vectors › Going Deeper: Advanced Triangle Trig & Vectors
Parent functions & transformations of $y=a\,f(bx-c)+d$: Parents: $c,\ x,\ |x|,\ \sqrt{x},\ x^2,\ x^3,\ \dfrac1x,\ \lfloor x\rfloor$ .; Course Review › Functions and Their Graphs
Partial fraction decomposition: For a proper rational $\dfrac{N(x)}{D(x)}$ (divide first if improper), split by the factors of $D$ :; Course Review › Systems, Matrices & Determinants
Partial fractions with repeated and irreducible factors: Two cases beyond the distinct-linear setup demand extra terms:; Systems, Matrices, and Determinants › Going Deeper: Advanced Systems & Matrices
Partial-fraction decomposition: To split a proper rational function $\dfrac{N(x)}{D(x)}$ (with $\deg N<\deg D$ ; otherwise divide first), fully factor $D$ and assign one term per factor: \[ $\frac{A}{x-a}$ , $\frac{A_1}{x-a}$ + $\frac{A_2}{(x-a)^2}$ +… (repeated), Bx+C…; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
Piecewise functions: A piecewise function uses different formulas on different parts of the domain.; Functions and Their Graphs › Piecewise Functions and Average Rate of Change
Piecewise functions: solving for TWO parameters: Suppose a piecewise $f$ has two unknown constants $a,b$ and you want it smooth (continuous and differentiable) at the split point $x=c$ .; Introduction to Limits › Going Deeper: Advanced Limit Ideas
Plotting and converting points: A polar point $(r,\theta)$ is located by rotating $\theta$ from the positive $x$ -axis, then moving $r$ along that ray (if $r<0$ , move backward, i.e. $180^\circ$ opposite).; Polar Coordinates and Complex Numbers › Polar Coordinates: Points and Conversions
Polar & rectangular conversion: Point $(r,\theta)$ .; Course Review › Polar Coordinates & Complex Numbers
Power sums and Newton's identities: Let $p_k=r_1^k+r_2^k+\cdots+r_n^k$ be the $k$ -th power sum of the roots and let $e_1,e_2,\dots$ be the elementary symmetric sums ( $e_1=\sum r_i,\ e_2=\sum_{i<j}r_ir_j,\ \dots$ ).; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
Product-to-Sum and Sum-to-Product: These identities trade a product for a sum (easier to integrate or evaluate) and vice versa: \[; Trigonometric Functions › Going Deeper: Advanced Trig Ideas
Rational functions $f(x)=\dfrac{N(x)}{D(x)}$: Course Review › Polynomial and Rational Functions
Reading $y=a\,f(bx-c)+d$: Starting from $y=f(x)$ :; Functions and Their Graphs › Transformations of Functions
Reading a graph: Functions and Their Graphs › Analyzing Graphs
Reading a proof: Do not read a proof the way you read a rule.; Sequences, Series, and the Binomial Theorem › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Introduction to Limits › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Functions and Their Graphs › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Polynomial and Rational Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Exponential and Logarithmic Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Trigonometric Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Analytic Trigonometry › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Triangle Trigonometry and Vectors › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Polar Coordinates and Complex Numbers › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Systems, Matrices, and Determinants › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Conic Sections and Parametric Equations › Why It Works
Recognize the family: Polar Coordinates and Complex Numbers › Graphs of Polar Equations
Reference Angle: The reference angle $\theta'$ is the acute angle between the terminal side and the $x$ -axis.; Trigonometric Functions › Reference Angles and Signs by Quadrant
Remainder modulo $(x-a)(x-b)$: Dividing $f(x)$ by a quadratic $(x-a)(x-b)$ leaves a remainder of degree $<2$ , so $r(x)=px+q$ is linear: \[ f(x)=(x-a)(x-b) q(x)+px+q.; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
Rewrite before you grind: Most precalc problems get easy after ONE rewrite: exponential equations → same base; log equations → exponent form ( $\log_b x = y \iff b^y = x$ ); rational expressions → factor first.; Core Ideas in Plain Terms › Problem-Solving Playbook
Root formula: Every nonzero complex number $z=r\,\text{cis}\,\theta$ has exactly $n$ distinct $n$ th roots: \[ z_k= $\sqrt[n]{r}$ cis ( $\frac{\theta+360^\circ k}{n}$ ), k=0,1,2,…,n-1.; Polar Coordinates and Complex Numbers › The $n$ th Roots of a Complex Number
Roots of unity: sums and products: The $n$ th roots of unity are the solutions of $z^n=1$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Row-echelon form: Write the system as an augmented matrix and use elementary row operations to reach row-echelon form (leading $1$ 's stepping to the right, zeros below), then back-substitute.; Systems, Matrices, and Determinants › Three-Variable Linear Systems: Gaussian Elimination
Sequences and factorial notation: A sequence is a function whose domain is the positive integers.; Sequences, Series, and the Binomial Theorem › Sequences: Explicit & Recursive Formulas, Factorials
Setting up applied problems: Bearings are measured clockwise from north (e.g. N $40^\circ$ E is $40^\circ$ east of due north).; Triangle Trigonometry and Vectors › Applications: Bearings, Forces, and Navigation
Sigma notation: The sum of terms $a_k$ from $k=1$ to $n$ is written \[ ∑_k=1^n a_k = a_1+a_2+…+a_n .; Sequences, Series, and the Binomial Theorem › Summation (Sigma) Notation & Its Properties
Sigma notation, induction, counting: Sigma: $\displaystyle\sum_{i=1}^{n}a_i=a_1+a_2+\cdots+a_n$ .; Course Review › Sequences, Series & the Binomial Theorem
Signs by Quadrant (“All Students Take Calculus”): Trigonometric Functions › Reference Angles and Signs by Quadrant
Sinusoidal Form: For $y=a\sin\big(b(x-c)\big)+d$ (and likewise for cosine) with $b>0$ :; Trigonometric Functions › Graphs of Sine and Cosine
Slant asymptotes and crossing the horizontal asymptote: Slant asymptote occurs exactly when $\deg N=\deg D+1$ : polynomial-divide $N$ by $D$ ; the quotient (a line $y=mx+b$ ) is the slant asymptote and the remainder term $\to0$ as $x\to\pm\infty$ .; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
SOH-CAH-TOA: For an acute angle $\theta$ in a right triangle with the opposite leg, adjacent leg, and hypotenuse: \[ θ=opphyp, θ=adjhyp, θ=oppadj.; Trigonometric Functions › Right-Triangle Trig and Fundamental Identities
Solving a linear recurrence for a closed form: A first-order recurrence $a_n=r\,a_{n-1}+c$ (with $r\ne 1$ ) can be turned into an explicit formula.; Sequences, Series, and the Binomial Theorem › Going Deeper: Advanced Sequences, Series & Binomial
Solving strategy: Exponential and Logarithmic Functions › Exponential and Logarithmic Equations
Solving systems & Gaussian elimination: Methods: substitution, elimination, or row-reduce the augmented matrix to row-echelon form (leading 1's, zeros below) then back-substitute.; Course Review › Systems, Matrices & Determinants
Special determinant identities and singular conditions: Some determinants have closed forms worth memorizing:; Systems, Matrices, and Determinants › Going Deeper: Advanced Systems & Matrices
Standard forms (center/vertex $(h,k)$ ): Circle: $(x-h)^2+(y-k)^2=r^2$ .; Course Review › Conic Sections & Parametric Equations
Standard forms of a hyperbola (center $(h,k)$ ): \[ $\frac{(x-h)^2}{a^2}$ - $\frac{(y-k)^2}{b^2}$ =1 (opens left/right), $\frac{(y-k)^2}{a^2}$ - $\frac{(x-h)^2}{b^2}$ =1 (opens up/down).; Conic Sections and Parametric Equations › Hyperbolas: Vertices, Foci, Asymptotes, Eccentricity
Standard forms of a parabola (vertex $(h,k)$ ): A parabola is the set of points equidistant from a fixed focus and a fixed line, the directrix.; Conic Sections and Parametric Equations › Parabolas: Focus, Directrix, and Latus Rectum
Standard forms of an ellipse (center $(h,k)$ , $a>b>0$ ): \[ $\frac{(x-h)^2}{a^2}$ + $\frac{(y-k)^2}{b^2}$ =1 (major axis horizontal), $\frac{(x-h)^2}{b^2}$ + $\frac{(y-k)^2}{a^2}$ =1 (major axis vertical).; Conic Sections and Parametric Equations › Ellipses: Center, Axes, Foci, Eccentricity
Standard models: \[ Growth/decay: A=A_0 e^kt (k>0 grows, k<0 decays) Half-life: A=A_0(12)^t/h \] \[ Compound: A=P(1+ $\tfrac{r}{n}$ )^nt Continuous: A=Pe^rt \] \[ Logistic: P(t)=c1+ae^-bt (limi…; Exponential and Logarithmic Functions › Modeling with Exponential and Logarithmic Functions
Stewart's Theorem: cevians, medians, and bisectors: A cevian is a segment from a vertex to a point on the opposite side.; Triangle Trigonometry and Vectors › Going Deeper: Advanced Triangle Trig & Vectors
Strategy for a proof: Work one side only (usually the messier one) until it matches the other.; Analytic Trigonometry › Verifying Identities
Substitution toolkit: Use $x=r\cos\theta$ , $y=r\sin\theta$ , and $r^2=x^2+y^2$ .; Polar Coordinates and Complex Numbers › Converting Equations Between Polar and Rectangular Form
Sum/difference, double, half, power-reducing, product-to-sum: Course Review › Analytic Trigonometry
Summing equally spaced sines and cosines (Dirichlet kernel): For an arithmetic progression of angles $a,\,a+d,\,a+2d,\dots$ , the telescoping idea from sum-to-product gives closed forms.; Analytic Trigonometry › Going Deeper: Advanced Analytic Trig
Systems of logarithmic equations: Treat the logs as unknowns.; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Telescoping products of cosines: Repeatedly using $\sin 2\alpha=2\sin\alpha\cos\alpha$ collapses a whole product of doubling-angle cosines into a single ratio of sines.; Analytic Trigonometry › Going Deeper: Advanced Analytic Trig
Telescoping sums of logs: A sum of logs telescopes through the quotient rule rather than change of base: \[ ∑_k=1^n_b $\frac{k+1}{k}$ =_b ( $\frac{2}{1}$ · $\frac{3}{2}$ … $\frac{n+1}{n}$ ) =_b(n+1).; Exponential and Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Telescoping sums via partial fractions: A sum telescopes when each term splits into a difference so that interior terms cancel in pairs.; Sequences, Series, and the Binomial Theorem › Going Deeper: Advanced Sequences, Series & Binomial
The 2×2 inverse: For $A =; Systems, Matrices, and Determinants › The Matrix Form $A\mathbf{x} = \mathbf{b}$
The ambiguous case as a parameter count: Solving a triangle means finding all $6$ parts from $3$ given ones.; Triangle Trigonometry and Vectors › Going Deeper: Advanced Triangle Trig & Vectors
The area problem: The other great problem of calculus is finding the area under a curve $y=f(x)$ over $[a,b]$ .; Introduction to Limits › The Tangent-Line Problem, the Derivative, and the Area Problem
The borderline cut: a parabola: Tilt the plane until it is exactly PARALLEL to the cone's slant side and the curve never closes up — that single borderline angle produces the parabola.; Conic Sections and Parametric Equations › Where Conics Come From: Slicing a Cone
The essentials: A complex number has the form $z=a+bi$ where $a$ is the real part, $b$ is the imaginary part, and $i^2=-1$ .; Polar Coordinates and Complex Numbers › Complex Numbers: Operations and the Complex Plane
The general second-degree equation: With no $xy$ term, $Ax^2+Cy^2+Dx+Ey+F=0$ classifies by comparing $A$ and $C$ :; Conic Sections and Parametric Equations › Classifying Conics from the General Form
The idea: A solution of a system is an ordered pair (or triple) that satisfies every equation at once.; Systems, Matrices, and Determinants › Solving Systems by Substitution and Elimination
The identities you build everything from: Reciprocal: $\csc\theta=\dfrac{1}{\sin\theta}$ , $\sec\theta=\dfrac{1}{\cos\theta}$ , $\cot\theta=\dfrac{1}{\tan\theta}$ .; Analytic Trigonometry › Fundamental Identities & Simplifying
The identity $z^n+z^{-n}=2\cos n\theta$: If $z=\text{cis}\,\theta$ lies on the unit circle then $\dfrac1z=z^{-1}=\text{cis}(-\theta)=\cos\theta-i\sin\theta$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
The limit laws: If $\lim\limits_{x\to a}f(x)$ and $\lim\limits_{x\to a}g(x)$ both exist, then limits distribute over the arithmetic: \[ _x→ a[f± g]= f± g, _x→ a[f· g]= f· g, \] \[ _x→ a $\frac{f}{g}$ =…; Introduction to Limits › Evaluating Limits Algebraically
The line–conic tangency condition: Substitute a line $y=mx+c$ into a conic and collect a quadratic in $x$ .; Conic Sections and Parametric Equations › Going Deeper: Advanced Conics & Parametrics
The number $e$: The natural base is the irrational constant \[ e=_n→∞(1+1n)^n≈ 2.71828 .; Exponential and Logarithmic Functions › The Natural Base $e$
The principle of induction: To prove a statement $P(n)$ holds for every positive integer $n$ :; Sequences, Series, and the Binomial Theorem › Mathematical Induction
The question changes from "what" to "when": Once a coefficient is a letter, "solve the system" splits into three questions: for which values of the parameter is there exactly one solution, for which is there none, and for which are there infinitely many?; Systems, Matrices, and Determinants › Systems With a Parameter
The six functions & unit circle: On the unit circle a terminal point is $(\cos\theta,\sin\theta)$ ; $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ , and $\csc,\sec,\cot$ are the reciprocals of $\sin,\cos,\tan$ .; Course Review › Trigonometric Functions
The three-part continuity test: A function $f$ is continuous at $x=a$ if all three hold: \[ (1) f(a) is defined, (2) _x→ af(x) exists, (3) _x→ af(x)=f(a).; Introduction to Limits › Continuity at a Point; Types of Discontinuity
Three counting tools: Sequences, Series, and the Binomial Theorem › Counting Principles: FCP, Permutations, Combinations
Three tools: Rational Zero Test: every rational zero of $f$ has the form $\dfrac{p}{q}$ , where $p\mid(\text{constant term})$ and $q\mid(\text{leading coefficient})$ .; Polynomial and Rational Functions › Finding Real Zeros: Rational Zero Test, Descartes, Bounds
Two area formulas: SAS formula (two sides and the included angle): \[ Area=12 ab C=12 bc A=12 ac B.; Triangle Trigonometry and Vectors › Area of a Triangle
Two useful forms: Standard (vertex) form: $f(x)=a(x-h)^2+k$ has vertex $(h,k)$ and axis of symmetry $x=h$ .; Polynomial and Rational Functions › Quadratic Functions
Vector basics: A vector in component form is $\vec u=\langle a,b\rangle$ .; Triangle Trigonometry and Vectors › Vectors: Component Form and Operations
Vector proofs: perpendicularity and right triangles: The dot product turns geometry into algebra.; Triangle Trigonometry and Vectors › Going Deeper: Advanced Triangle Trig & Vectors
Vectors, dot product & projection: $\mathbf{v}=\langle v_1,v_2\rangle$ , magnitude $\|\mathbf{v}\|=\sqrt{v_1^2+v_2^2}$ , direction angle $\theta$ with $\mathbf{v}=\|\mathbf{v}\|\langle\cos\theta,\sin\theta\rangle$ .; Course Review › Triangle Trigonometry & Vectors
Vieta's formulas: coefficients from roots: If $x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0$ (monic) has roots $r_1,\dots,r_n$ , then the elementary symmetric sums of the roots are the coefficients (up to sign): \[ ∑ r_i=-c_n-1, ∑_i<j r_ir_j=c_n-2, …, r_1r_2… r_n=(-1)^n c_0.; Polynomial and Rational Functions › Going Deeper: Advanced Polynomial & Rational Ideas
Weighted geometric sums: $\sum k\,r^{\,k}$: The plain geometric sum has a closed form; multiplying each term by $k$ needs one extra idea.; Sequences, Series, and the Binomial Theorem › Going Deeper: Advanced Sequences, Series & Binomial
What a limit means: The limit of $f(x)$ as $x$ approaches $a$ is the single value $L$ that the outputs $f(x)$ get arbitrarily close to as $x$ gets close to $a$ (from both sides), without necessarily equaling $a$ .; Introduction to Limits › The Idea of a Limit; Estimating Numerically and Graphically
What happens as $x\to\pm\infty$: $\displaystyle\lim_{x\to\infty}f(x)=L$ means the outputs level off at the height $y=L$ as $x$ grows large; the line $y=L$ is a horizontal asymptote.; Introduction to Limits › Limits at Infinity and End Behavior
What is a function?: A function $f$ assigns to each input $x$ (from the domain) exactly one output $y=f(x)$ .; Functions and Their Graphs › Function Notation, Domain & Range, Evaluating
Why this belongs here: Splitting $\dfrac{5x - 1}{(x-1)(x+2)}$ into $\dfrac{A}{x-1} + \dfrac{B}{x+2}$ means finding numbers $A$ and $B$ that make two polynomials identical.; Systems, Matrices, and Determinants › Partial Fractions Are a System in Disguise