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Every rule and concept in Algebra 2, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (215)

“Less thAND” and “greatOR”: After isolating the absolute value (with $c>0$ ):; Equations, Inequalities & Absolute Value › Absolute Value Inequalities
“Let $\log_2 3 = a$ ” — Expressing Logs Symbolically: When a problem fixes one log as a letter, express others by breaking numbers into prime factors and applying the three laws.; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
$n$ th Roots & Rational Exponents: $\sqrt[n]{a}=a^{1/n}$ , $a^{m/n}=\sqrt[n]{a^m}=\big(\sqrt[n]{a}\big)^m$ .; Course Review › Radical Functions & Rational Exponents
A factoring toolkit: Polynomial Functions › Factoring Higher-Degree Polynomials
A Fraction Inside a Fraction: A complex fraction has fractions in its numerator, denominator, or both.; Rational Expressions & Functions › Complex Fractions
A function is an input-output machine: $f(x)$ means "the output when the input is $x$ ." Small changes to the formula move the graph in predictable ways: $f(x) + k$ shifts it up, $f(x - h)$ shifts it right, and a minus sign flips it.; Core Ideas in Plain Terms › Functions and Their Graphs
A Logarithm is an Exponent: A logarithm answers the question “to what power?” The definition is the inverse relationship \[ _b y = x b^x = y, b>0, b≠ 1, y>0.; Exponential & Logarithmic Functions › Logarithms: Definition & Evaluating
A new number: The imaginary unit is defined by $i=\sqrt{-1}$ , so that $i^2=-1$ .; Quadratic Functions & Complex Numbers › Complex Numbers: the Imaginary Unit $i$
Add, subtract, multiply, divide: Add/subtract: combine real parts and imaginary parts separately.; Quadratic Functions & Complex Numbers › Operations with Complex Numbers
Add, Subtract, Multiply, Divide: Radical Functions & Rational Exponents › Operations with Radicals
Anatomy of a parabola: For $y=ax^2+bx+c$ :; Quadratic Functions & Complex Numbers › Graphing Parabolas
Anatomy of an Exponential: An exponential function has the form \[ y = ab^x, a ≠ 0, b > 0, b ≠ 1.; Exponential & Logarithmic Functions › Exponential Functions $y = ab^x$
Arithmetic (common difference $d$ ): \[ a_n=a_1+(n-1)d, S_n= $\frac{n}{2}$ (a_1+a_n)= $\frac{n}{2}$ (2a_1+(n-1)d).; Course Review › Sequences & Series
Asymptotes & Holes: For $f(x)=\dfrac{p(x)}{q(x)}$ in lowest terms:; Course Review › Rational Expressions & Functions
Asymptotes, plainly: A rational function blows up where its denominator hits $0$ (vertical asymptotes), and for huge $x$ it flattens toward the ratio of the leading terms (horizontal asymptote).; Core Ideas in Plain Terms › Going Deeper: Polynomials and Rational Functions
Bayes' Theorem: Reversing the Condition: Conditional probabilities are not symmetric: $P(A \mid B) \ne P(B \mid A)$ in general.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Big idea: A conic section is a curve you get by slicing a double cone with a flat plane.; Conic Sections › The Four Conics & Review Formulas
Big idea: A relation is any set of ordered pairs $(x,y)$ .; Functions, Relations & Transformations › Relations, Functions & the Vertical Line Test
Binomial Probability: For $n$ independent trials, success probability $p$ , exactly $k$ successes: \[ P(k)= $\binom{n}{k}$ p^k(1-p)^ n-k.; Course Review › Probability & Statistics
Binomial Threshold Probabilities (“At Least $k$ ”): For a binomial with $n$ trials and success probability $p$ , a threshold question sums several exact- $k$ terms: \[ P(X ≥ k) = ∑_j=k^n $\binom{n}{j}$ p^j(1-p)^n-j.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Building a conic from conditions: So far you have read a conic off its equation.; Conic Sections › Going Deeper: Advanced Conics
Building a perfect square: To complete the square on $x^2+bx$ , add $\left(\dfrac{b}{2}\right)^2$ .; Quadratic Functions & Complex Numbers › Completing the Square
Building a polynomial from finite differences: Tabulate a degree- $n$ polynomial at equally spaced inputs and take successive differences of the outputs.; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Building a quadratic from transformed roots: To construct a monic quadratic with prescribed roots $s_1,s_2$ , use \[ x^2-(s_1+s_2) x+s_1 s_2=0.; Quadratic Functions & Complex Numbers › Going Deeper: Advanced Quadratics & Complex Numbers
Center, Then Spread: Center: the mean $\bar{x} = \dfrac{\sum x}{n}$ , the median (middle value when ordered), and the mode (most frequent).; Probability & Statistics › Measures of Center and Spread
Change of Base: To evaluate a log in any base with a calculator: \[ _b M = $\frac{\log M}{\log b}$ = $\frac{\ln M}{\ln b}$ .; Exponential & Logarithmic Functions › Properties of Logarithms
Change-of-Base Chains & Telescoping Products: Change of base can be written in any common base $c$ : \[ _b M = $\frac{\log_c M}{\log_c b}$ .; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Choosing the Right Model: Real-world patterns are often sequences or series.; Sequences & Series › Applications
Circle & Parabola: Circle: center $(h,k)$ , radius $r$ : \[ (x-h)^2+(y-k)^2=r^2.; Course Review › Conic Sections
Classifying the full second-degree equation via $B^2-4AC$: When an $xy$ term is present the conic is rotated, and reading $A$ vs. $C$ is no longer enough.; Conic Sections › Going Deeper: Advanced Conics
Clear the Denominators: To solve (not simplify) a rational equation, multiply every term by the LCD to clear all fractions, then solve the resulting polynomial equation.; Rational Expressions & Functions › Solving Rational Equations & Extraneous Solutions
Combining functions: For two functions $f$ and $g$ : \[ (f+g)(x)=f(x)+g(x), (f-g)(x)=f(x)-g(x), \] \[ (fg)(x)=f(x)· g(x), ( $\tfrac{f}{g}$ )(x)= $\frac{f(x)}{g(x)}$ (g(x)≠ 0).; Functions, Relations & Transformations › Operations on Functions
Combining polynomials: Add / subtract: combine like terms.; Polynomial Functions › Adding, Subtracting, and Multiplying Polynomials
Combining Two Groups' Means and Variances: To pool two groups of sizes $n_1, n_2$ , the combined mean is the weighted average — never the plain average of the two means unless the groups are the same size: \[ x_combined = n_1x_1 + n_2x_2n_1 + n_2.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Common and Natural Logs: Two bases are so useful they get their own notation:; Exponential & Logarithmic Functions › Logarithms: Definition & Evaluating
Common Denominators: You can only combine fractions over a common denominator.; Rational Expressions & Functions › Adding & Subtracting (LCD)
Common Difference and the $n$ th Term: A sequence is arithmetic when each term differs from the one before it by the same amount, the common difference $d$ .; Sequences & Series › Arithmetic Sequences
Common Ratio and the $n$ th Term: A sequence is geometric when each term is a fixed multiple of the one before it, the common ratio $r$ .; Sequences & Series › Geometric Sequences
Complementary Counting and “At Least One”: When a direct count splits into many cases, count the opposite and subtract from the total.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Complex Numbers: $i=\sqrt{-1}$ , $i^2=-1$ .; Course Review › Quadratic Functions & Complex Numbers
Composing several transformations into one mapping: A chain of transformations can always be collapsed into a single rule $y=a\,f\bigl(b(x-h)\bigr)+k$ .; Functions, Relations & Transformations › Going Deeper: Advanced Function Ideas
Composition: The composition $(f\circ g)(x)=f\bigl(g(x)\bigr)$ means: run $g$ first, then feed its output into $f$ .; Functions, Relations & Transformations › Composition of Functions
Continued Fractions: A continued fraction nests division inside division: \[ a_0+1a_1+1a_2+1a_3+….; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
Continuous Compounding & Logistic Growth: Two continuous models worth knowing beyond $A = Pe^{rt}$ :; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Core Moves: Equations: isolate the variable by undoing operations in reverse order — distribute, clear fractions with the LCD, gather variable terms on one side, divide by the coefficient.; Course Review › Equations, Inequalities & Absolute Value
Counting Successes in Repeated Trials: A binomial experiment has: (1) a fixed number of trials $n$ , (2) two outcomes per trial — success or failure, (3) a constant success probability $p$ , and (4) independent trials.; Probability & Statistics › Binomial Probability (Introduction)
Counting with Restrictions: Circular & Identical Items: Not every arrangement is a plain permutation — watch for two twists.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Counting zeros: Fundamental Theorem of Algebra: every polynomial of degree $n\ge1$ has exactly $n$ zeros in the complex numbers, counted with multiplicity.; Polynomial Functions › Fundamental Theorem of Algebra
Counting: Permutations & Combinations: Fundamental Counting Principle: multiply the choices at each stage.; Course Review › Probability & Statistics
Cramer's Rule for a $2\times2$ system: For $; Linear Systems & Matrices › Solving Systems with Determinants: Cramer's Rule
Definitions: $y=b^x$ ( $b>0,\ b\ne1$ ): growth if $b>1$ , decay if $0<b<1$ ; horizontal asymptote $y=0$ .; Course Review › Exponential & Logarithmic Functions
Degenerate conics: A “conic” equation does not always draw a smooth curve.; Conic Sections › Going Deeper: Advanced Conics
Denesting a Nested Radical $\sqrt{a+\sqrt{b}}$: Sometimes a radical hidden inside another radical can be “unpacked” into a sum of two simpler square roots.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
Determinant properties and special identities: These rules let you evaluate determinants without brute force:; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Determinants, Inverse & Cramer's Rule: $2\times2$ : $; Course Review › Linear Systems & Matrices
Dimensions, equality, and the simple operations: A matrix is a rectangular array of numbers.; Linear Systems & Matrices › Matrix Basics
Discrete vs. continuous: A discrete function is defined only at separated points (draw dots), e.g. “number of students” vs. “number of pizzas.” A continuous function is defined on an unbroken interval (draw a solid curve), e.g. “temperature” vs. “time.”; Functions, Relations & Transformations › Function Notation, Evaluating, Discrete vs. Continuous
Discriminant & parameter conditions: Treat the discriminant $D=b^2-4ac$ as a function of an unknown parameter to force a desired root behavior:; Quadratic Functions & Complex Numbers › Going Deeper: Advanced Quadratics & Complex Numbers
Distance from Zero: $|x|$ is the distance of $x$ from $0$ , so it is never negative.; Equations, Inequalities & Absolute Value › Absolute Value Equations
Division & Key Theorems: Synthetic division divides by $(x-c)$ : bring down, multiply by $c$ , add.; Course Review › Polynomial Functions
Domain & Range of Radical Functions and Their Inverses: For a radical function, the domain of the function is the range of its inverse, and vice versa — reflecting across $y=x$ swaps the two axes.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
Ellipse & Hyperbola (center $(h,k)$ ): Ellipse ( $a>b$ , $a$ = semi-major): $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ , foci on major axis with $c^2=a^2-b^2$ .; Course Review › Conic Sections
End Behavior (Leading Term Test): For degree $n$ with leading coefficient $a$ :; Course Review › Polynomial Functions
Euler's Number $e$: The constant $e \approx 2.71828$ is the natural base of growth.; Exponential & Logarithmic Functions › The Number $e$ & Continuous Growth
Evaluating by dividing: Remainder Theorem: When $f(x)$ is divided by $x-c$ , the remainder equals $f(c)$ .; Polynomial Functions › The Remainder and Factor Theorems
Expected Value: The Long-Run Average: The expected value of a numerical outcome $X$ weights each value by its probability: \[ E(X) = ∑ x_i P(x_i).; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Exploiting even and odd symmetry: A function is even if $f(-x)=f(x)$ (graph symmetric across the $y$ -axis) and odd if $f(-x)=-f(x)$ (symmetric through the origin).; Functions, Relations & Transformations › Going Deeper: Advanced Function Ideas
Exponential Equations Hidden as Quadratics: An equation with terms in $2^x$ and $2^{2x}$ (or $e^x$ and $e^{2x}$ ) is secretly a quadratic.; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Find $x+y+z$ without solving the whole system: Sometimes a problem asks only for a combination of the unknowns — not each one.; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Focus and directrix (intro): Write an upward parabola as $y=\dfrac{1}{4p}(x-h)^2+k$ .; Conic Sections › Parabolas as Conics
From words to equations: To model a word problem:; Linear Systems & Matrices › Applications: Setting Up Systems
Function notation: We write $f(x)$ (read “ $f$ of $x$ ”) for the output of function $f$ at input $x$ .; Functions, Relations & Transformations › Function Notation, Evaluating, Discrete vs. Continuous
Functional equations: solve for the rule, not a number: A functional equation constrains an unknown function $f$ instead of an unknown number.; Functions, Relations & Transformations › Going Deeper: Advanced Function Ideas
Functions & Domain/Range: A relation pairs inputs with outputs; a function assigns exactly one output to each input (passes the vertical line test).; Course Review › Functions, Relations & Transformations
Geometric (common ratio $r$ ): \[ a_n=a_1 r^ n-1, S_n=a_11-r^ n1-r (r≠1).; Course Review › Sequences & Series
Growth, Decay & Interest: Course Review › Exponential & Logarithmic Functions
Holes vs. Vertical Asymptotes — Told Precisely: Both come from a zero of the original denominator, but they behave very differently.; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
How many solutions?: Compare the two lines:; Linear Systems & Matrices › Solving Two-Variable Systems (Review)
i is the square root of −1: Some equations have no real solution because you'd need the square root of a negative.; Core Ideas in Plain Terms › Complex Numbers
Identifying a Conic from $Ax^2+Cy^2+\cdots=0$: Course Review › Conic Sections
Inequalities Solved by Substitution: When an inequality contains a repeated chunk, let a new variable stand for it.; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
Infinite Geometric Series in Disguise: Many word problems hide a convergent geometric series.; Sequences & Series › Going Deeper: Advanced Sequences & Series
Interest Formulas: For principal $P$ , annual rate $r$ (as a decimal), and time $t$ in years: \[ compounded n times/year: A = P(1 + $\frac{r}{n}$ )^nt, compounded continuously: A = P e^rt.; Exponential & Logarithmic Functions › The Number $e$ & Continuous Growth
Interval Notation: Course Review › Equations, Inequalities & Absolute Value
Invertibility via monotonicity: A function is invertible exactly when it is one-to-one (passes the horizontal line test).; Functions, Relations & Transformations › Going Deeper: Advanced Function Ideas
Isolate, Power, Check: To solve, isolate the radical, then raise both sides to the matching power to undo it.; Radical Functions & Rational Exponents › Solving Radical & Rational-Exponent Equations
Iterated composition and involutions: Composing $f$ with itself is written $f^{[n]}=\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ copies}}$ , so $f^{[2]}(x)=f(f(x))$ .; Functions, Relations & Transformations › Going Deeper: Advanced Function Ideas
Keep it Balanced: An equation is a balance scale.; Equations, Inequalities & Absolute Value › Solving Multi-Step Linear Equations
Latus rectum, focal chords, and the reflection property: A focal chord is any chord through a focus.; Conic Sections › Going Deeper: Advanced Conics
Linear Transformations of Data: $y = ax + b$: Shifting and scaling every data value changes center and spread in predictable ways.; Probability & Statistics › Going Deeper: Advanced Probability & Statistics
Log Properties & Change of Base: Course Review › Exponential & Logarithmic Functions
Logarithmic & Exponential Inequalities: Solving an inequality adds a direction rule on top of the equation techniques:; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Logs undo exponents: $\log_b x = y$ is just another way to write $b^y = x$ — it asks "what power turns $b$ into $x$ ?" That is why logs are the tool for solving equations where the unknown is stuck in an exponent, and why they turn multiplication into addition.; Core Ideas in Plain Terms › Exponentials and Logarithms
Mathematical Induction: Induction proves a statement $P(n)$ is true for every whole number $n \ge 1$ using two steps, like knocking over an infinite line of dominoes:; Sequences & Series › Going Deeper: Advanced Sequences & Series
Matrix Operations: Add/subtract entrywise (same dimensions).; Course Review › Linear Systems & Matrices
Matrix powers $A^n$ and recurrences: Repeated multiplication $A^n = A\cdot A \cdots A$ ( $n$ factors) models step-by-step processes (Markov chains, Fibonacci-type recurrences).; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Mean, Standard Deviation & the Normal Curve: \[ mean x= $\frac{\sum x}{n}$ , standard deviation σ=∑ (x-x)^2n.; Course Review › Probability & Statistics
Models You Will See: Exponential & Logarithmic Functions › Applications
Modulus & argument of a complex number: Plot $z=a+bi$ as the point $(a,b)$ .; Quadratic Functions & Complex Numbers › Going Deeper: Advanced Quadratics & Complex Numbers
Multiply Straight Across; Divide by Flipping: Multiply: factor everything, cancel any factor in a numerator against a matching factor in a denominator, then write what remains.; Rational Expressions & Functions › Multiplying & Dividing
Multiply the Choices: If a task is made of several independent stages, and stage $1$ can happen in $m$ ways, stage $2$ in $n$ ways, and so on, then the whole task can happen in \[ m × n × … ways. \] This is the engine behind every counting formula in this…; Probability & Statistics › The Fundamental Counting Principle
Nested Absolute Values: Peel One Layer at a Time: An equation like $\big|\,|x-2|-3\,\big| = 4$ has an absolute value inside another.; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
Nonlinear systems: line–conic and conic–conic: When at least one equation is a conic (circle, parabola, ellipse, hyperbola), count solutions geometrically:; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Operations: Course Review › Rational Expressions & Functions
Order Matters vs. Order Doesn't: Both count ways to choose $r$ items from $n$ distinct items.; Probability & Statistics › Permutations and Combinations
Parameter Conditions for a Solution Set: Sometimes the unknown is a parameter that controls whether — and how many — solutions exist.; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
Parent Functions: Course Review › Functions, Relations & Transformations
Parent Graphs and Their Domains: Radical Functions & Rational Exponents › Graphing Square-Root & Cube-Root Functions
Partial-Fraction Decomposition: The reverse of adding fractions: split one complicated rational expression into a sum of simple ones.; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
Power sums and Newton's identities: Let $p_k=r_1^k+\cdots+r_n^k$ be the $k$ th power sum of the roots and let $e_k$ be the $k$ th elementary symmetric function (so $e_1=\sum r_i$ , $e_2=\sum_{i<j} r_ir_j$ , …).; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Power Sums and the Cube Identity: Three closed forms let you sum powers of the first $n$ whole numbers without adding term by term:; Sequences & Series › Going Deeper: Advanced Sequences & Series
Probability Rules: $P(E)=\dfrac{\text{favorable}}{\text{total}}$ , with $0\le P(E)\le1$ ; complement $P(E')=1-P(E)$ .; Course Review › Probability & Statistics
Putting it all together: To sketch $y=f(x)$ :; Polynomial Functions › Graphing Polynomial Functions
Radical Equations & Inverses: Solve: isolate the radical, raise both sides to the index power, solve, then check for extraneous roots (squaring can introduce false solutions).; Course Review › Radical Functions & Rational Exponents
Rates Add: If one worker finishes a job in $a$ hours, their rate is $\dfrac{1}{a}$ job per hour.; Rational Expressions & Functions › Applications: Work/Rate & Mixture
Rational Root Theorem with bounds and Descartes' Rule: Three tools sharply narrow the candidate list before you test anything.; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Rational Root Theorem, multiplicity, and depressing: Rational Root Theorem: any rational zero of a polynomial with integer coefficients has the form $\dfrac{p}{q}$ , where $p$ divides the constant term and $q$ divides the leading coefficient.; Polynomial Functions › Finding All Zeros
Rational-Exponent Equations That Hide a Quadratic: An equation like $x^{2/3}-x^{1/3}-6=0$ looks exotic, but the exponents are in a $2:1$ ratio, so a $u$ -substitution turns it into a plain quadratic.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
Rationalizing Multi-Term Denominators: A two-term denominator clears with its conjugate.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
Reading a piecewise rule: A piecewise function uses different formulas on different parts of the domain.; Functions, Relations & Transformations › Piecewise Functions
Reading a proof: Do not read a proof the way you read a rule.; Conic Sections › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Probability & Statistics › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Equations, Inequalities & Absolute Value › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Functions, Relations & Transformations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Linear Systems & Matrices › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Quadratic Functions & Complex Numbers › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Polynomial Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Rational Expressions & Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Radical Functions & Rational Exponents › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Exponential & Logarithmic Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Sequences & Series › Why It Works
Reading and Writing $\Sigma$: The Greek capital sigma $\sum$ is shorthand for “add these up.” \[ ∑_n=1^k a_n = a_1 + a_2 + a_3 + … + a_k \] The letter $n$ is the index; it starts at the number written below $\sum$ and climbs by $1$ up to the number written above.; Sequences & Series › Summation (Sigma) Notation
Reading the transformation: Starting from a parent $f(x)$ , the graph of $y=a\,f(x-h)+k$ is transformed as follows:; Functions, Relations & Transformations › Transformations: $y=a\,f(x-h)+k$
Remainder of $P(x)$ modulo $(x-a)(x-b)$: Dividing $P(x)$ by a quadratic leaves a remainder of degree at most $1$ , say $R(x)=mx+k$ .; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Roots in arithmetic or geometric progression: When a problem promises the roots follow a pattern, choose symmetric variables so Vieta collapses nicely.; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Roots Undo Powers: Radical functions are the inverses of power functions.; Radical Functions & Rational Exponents › Inverses & the Root–Power Connection
Same Rules, Fractional Exponents: Every exponent law you know still holds — just add, subtract, or multiply the fractions: \[ x^ax^b=x^a+b, x^ax^b=x^a-b, (x^a)^b=x^ab, (xy)^a=x^ay^a, x^-a=1x^a.; Radical Functions & Rational Exponents › Properties of Exponents with Rational Powers
Shortest distance from a point to a conic: The closest point $Q$ on a conic to an external point $P$ has a clean geometric signature: the segment $PQ$ is perpendicular to the tangent line at $Q$ (it lies along the normal).; Conic Sections › Going Deeper: Advanced Conics
Sigma (Summation) Notation: \[ ∑_k=1^n a_k = a_1+a_2+…+a_n.; Course Review › Sequences & Series
Singular matrices: when there is no inverse: A square matrix $A$ is singular (non-invertible) exactly when $\det A = 0$ .; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Slant (Oblique) Asymptotes via Long Division: When $\deg P = \deg Q + 1$ (numerator exactly one degree higher), there is no horizontal asymptote — instead the graph follows a slant line $y=mx+b$ .; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
Solving for a Variable: A literal equation contains several letters.; Equations, Inequalities & Absolute Value › Literal Equations & Formulas
Solving Inequalities: Solve an inequality just like an equation, with one crucial rule: when you multiply or divide by a negative number, flip the inequality sign.; Equations, Inequalities & Absolute Value › Linear Inequalities & Interval Notation
Solving Methods & Discriminant: Course Review › Quadratic Functions & Complex Numbers
Solving Systems: Course Review › Linear Systems & Matrices
Standard equation of a circle: A circle with center $(h,k)$ and radius $r$ is the set of all points a distance $r$ from the center: \[ (x-h)^2+(y-k)^2=r^2.; Conic Sections › Circles
Standard equation of a hyperbola (center at origin): \[ $\dfrac{x^2}{a^2}$ - $\dfrac{y^2}{b^2}$ =1 (opens left–right), $\dfrac{y^2}{a^2}$ - $\dfrac{x^2}{b^2}$ =1 (opens up–down).; Conic Sections › Hyperbolas
Standard equation of an ellipse (center at origin): \[ $\dfrac{x^2}{a^2}$ + $\dfrac{y^2}{b^2}$ =1, a>b>0.; Conic Sections › Ellipses
Strategy & the Domain Check: Exponential & Logarithmic Functions › Solving Logarithmic Equations
Substitute to simplify: A scary equation is often a familiar one in disguise.; Core Ideas in Plain Terms › Problem-Solving Playbook
Substitution is the general tool: When one equation is a circle, a parabola or a hyperbola, elimination usually has nothing to cancel.; Linear Systems & Matrices › Systems That Are Not Lines
Summed Absolute Values: Full Casework by Critical Points: To solve $|x-a| + |x-b| = c$ you cannot square or split into two — instead find the critical points where each expression changes sign ( $x=a$ and $x=b$ ).; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
Summing a Finite Geometric Sequence: The sum of the first $n$ terms of a geometric sequence (with $r \ne 1$ ) is \[ S_n = a_1 1 - r^ n1 - r \]; Sequences & Series › Geometric Series (Finite Sums)
Summing an Arithmetic Sequence: A series is the sum of the terms of a sequence.; Sequences & Series › Arithmetic Series (Partial Sums)
Symmetric systems and elementary symmetric functions: A system is symmetric if swapping $x \leftrightarrow y$ leaves it unchanged, e.g.; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
Systems of Absolute-Value Inequalities Define Regions: A single absolute-value inequality in $x$ and $y$ carves the plane into a shape; a system intersects several such shapes.; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
Systems of Logarithmic Equations: A system mixing sums and products of logs is solved by condensing each equation, then converting to ordinary algebra.; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
Telescoping Sums: A sum telescopes when each term splits into two pieces that cancel against neighbors, leaving only the very first and very last.; Sequences & Series › Going Deeper: Advanced Sequences & Series
Telescoping Sums from Partial Fractions: Partial fractions turn many “impossible” sums into telescoping sums, where adjacent terms cancel in a cascade.; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
The $x^{\log x}$ Type — Take a Log of a Log-Exponent: When the variable appears in both the base and the exponent, as in $x^{\log x}$ , take $\log$ of both sides first; the power rule turns the exponent into a factor and creates a quadratic in $\log x$ .; Exponential & Logarithmic Functions › Going Deeper: Advanced Exponential & Log Ideas
The $x+\tfrac1x$ Family: Many problems become easy once you let $s=x+\dfrac1x$ .; Rational Expressions & Functions › Going Deeper: Advanced Rational Functions
The Addition Rule: For the probability that $A$ or $B$ happens: \[ P(A or B) = P(A) + P(B) - P(A and B).; Probability & Statistics › Mutually Exclusive & Overlapping Events (Addition Rule)
The Bell Curve: Many natural measurements (heights, test scores, errors) follow a normal distribution: a symmetric, bell-shaped curve centered at the mean $\mu$ , with spread set by $\sigma$ .; Probability & Statistics › The Normal Distribution, the Empirical Rule & z-Scores
The classification test: Given $Ax^2+Cy^2+Dx+Ey+F=0$ (no $xy$ term), look only at $A$ and $C$ :; Conic Sections › Identifying a Conic from General Form
The determinant of a square matrix: For a $2\times2$ matrix, the determinant is \[; Linear Systems & Matrices › Determinants: $2\times2$ and $3\times3$
The Distance Interpretation of $|x-a|+|x-b|$: Read $|x-a|$ as “the distance from $x$ to $a$ .” Then $|x-a| + |x-b|$ is the total distance from $x$ to the two fixed points $a$ and $b$ .; Equations, Inequalities & Absolute Value › Going Deeper: Advanced Ideas
The formula that always works: For $ax^2+bx+c=0$ with $a\neq0$ , \[ x=-b± $\sqrt{b^2-4ac}$ 2a.; Quadratic Functions & Complex Numbers › The Quadratic Formula and the Discriminant
The Full Checklist: For $f(x)=\dfrac{P(x)}{Q(x)}$ in lowest terms:; Rational Expressions & Functions › Graphing Rational Functions
The inverse of a $3\times3$ matrix: For a square matrix $A$ , the inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$ .; Linear Systems & Matrices › Going Deeper: Advanced Systems & Matrices
The Inverse of an Exponential: Because $\log_b x$ undoes $b^x$ , the graph of $y = \log_b x$ is the reflection of $y = b^x$ across the line $y = x$ .; Exponential & Logarithmic Functions › Graphs of Logarithmic Functions
The language of polynomials: A polynomial in one variable is a sum of terms of the form $a_n x^n$ , where each exponent is a whole number.; Polynomial Functions › Polynomial Vocabulary
The line–conic tangency condition (discriminant $=0$ ): To find where a line meets a conic, substitute the line into the conic.; Conic Sections › Going Deeper: Advanced Conics
The Master Conversion: A fractional exponent is a compact radical.; Radical Functions & Rational Exponents › Rational Exponents: Radical $\leftrightarrow$ Exponent Form
The matrix is just the equations with the letters left out: Writing $;2x + 3y = 7$ , $;x - y = 1;$ as $[; Linear Systems & Matrices › Row Reduction Without the Mystery
The Multiplication Rule: For the probability that both $A$ and $B$ happen: \[ P(A and B) = P(A)· P(B A).; Probability & Statistics › Independent & Dependent Events; Conditional Probability
The only three legal moves: Swap two rows — writing the equations in a different order. Scale a row by a nonzero constant — multiplying an equation through. Add a multiple of one row to another — the elimination step. Each is reversible, so none of them can gain or lose a …; Linear Systems & Matrices › Row Reduction Without the Mystery
The remainder theorem in one line: Plugging $a$ into a polynomial gives the remainder when dividing by $(x - a)$ : $P(a)$ is the remainder.; Core Ideas in Plain Terms › Going Deeper: Polynomials and Rational Functions
The same parabola, written three ways: Every quadratic function can be written in three equivalent forms.; Quadratic Functions & Complex Numbers › Three Forms of a Quadratic
The six parents: Each family of functions is built from a simple “parent.” Memorize their shapes, domains, and ranges.; Functions, Relations & Transformations › Parent Functions and Their Graphs
The solution is a region, not a point: For a system of linear inequalities, the solution set is the region where all shaded half-planes overlap.; Linear Systems & Matrices › Systems of Linear Inequalities
The strategy: A system in $x,y,z$ needs three equations.; Linear Systems & Matrices › Solving Three-Variable Systems by Elimination
The strategy is always "reduce the count": A $3\times3$ system is solved by turning it into a $2\times2$ system. 1.; Linear Systems & Matrices › Three Equations, Three Unknowns
The Symmetric Family $x^{1/2}+x^{-1/2}$: Expressions built from $\sqrt{x}$ and its reciprocal are symmetric: swapping $\sqrt{x}\leftrightarrow\frac{1}{\sqrt{x}}$ leaves them unchanged.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
The Three Laws (same base $b$ ): Exponential & Logarithmic Functions › Properties of Logarithms
The three methods: A system of two linear equations asks for the point $(x,y)$ that satisfies both equations at once.; Linear Systems & Matrices › Solving Two-Variable Systems (Review)
The Vertical Line Test: A graph represents a function if and only if no vertical line crosses it more than once.; Functions, Relations & Transformations › Relations, Functions & the Vertical Line Test
Theoretical, Experimental, Complement: For equally likely outcomes, \[ P(event) = number of favorable outcomestotal number of outcomes, 0 ≤ P ≤ 1.; Probability & Statistics › Basic Probability
Three Forms of a Quadratic: Course Review › Quadratic Functions & Complex Numbers
Three Variation Models: Every variation problem hides a constant $k$ .; Rational Expressions & Functions › Direct, Inverse & Joint Variation
Transformations of $y = b^x$: Starting from the parent $y = b^x$ , the function $y = a\,b^{\,x-h} + k$ is:; Exponential & Logarithmic Functions › Exponential Functions $y = ab^x$
Transformations of $y=a\,f\big(b(x-h)\big)+k$: Course Review › Functions, Relations & Transformations
Translating Words: Turn the words into an equation or inequality, solve, then interpret the answer in context.; Equations, Inequalities & Absolute Value › Applications & Word Problems
Two quick methods: Factoring uses the Zero Product Property: if $A\cdot B=0$ then $A=0$ or $B=0$ .; Quadratic Functions & Complex Numbers › Solving by Factoring and the Square-Root Method
Two Radicals: Isolate, Square, Square Again: When two radicals appear, you cannot clear both in one squaring.; Radical Functions & Rational Exponents › Going Deeper: Advanced Radicals & Rational Exponents
Two Statements Joined: A compound inequality joins two inequalities.; Equations, Inequalities & Absolute Value › Compound Inequalities (AND / OR)
Two Strategies: Exponential & Logarithmic Functions › Solving Exponential Equations
Two tools you already own (review): For points $(x_1,y_1)$ and $(x_2,y_2)$ : \[ Distance: d= $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ , Midpoint: M=( $\dfrac{x_1+x_2}{2}$ , $\dfrac{y_1+y_2}{2}$ ).; Conic Sections › The Four Conics & Review Formulas
Two ways to divide: Long division works for any divisor.; Polynomial Functions › Long Division and Synthetic Division
Variation: Course Review › Rational Expressions & Functions
Vertex form and direction of opening: A parabola is every point equidistant from a fixed focus and a fixed line called the directrix.; Conic Sections › Parabolas as Conics
Vieta's formulas: roots to coefficients: For a monic polynomial with roots $r_1,\dots,r_n$ (repeated by multiplicity), expanding $\prod (x-r_i)$ shows that the elementary symmetric functions of the roots are the coefficients, with alternating signs.; Polynomial Functions › Going Deeper: Advanced Polynomial Ideas
Vieta's formulas: roots without solving: For $ax^2+bx+c=0$ with roots $r_1,r_2$ , dividing by $a$ gives $x^2+\dfrac{b}{a}x+\dfrac{c}{a}=0$ .; Quadratic Functions & Complex Numbers › Going Deeper: Advanced Quadratics & Complex Numbers
What an $n$ th Root Means: $\sqrt[n]{a} = b$ exactly when $b^n = a$ .; Radical Functions & Rational Exponents › $n$ th Roots & Simplifying Radicals
What an inverse does: The inverse $f^{-1}$ undoes $f$ : it swaps inputs and outputs, so if $f(a)=b$ then $f^{-1}(b)=a$ .; Functions, Relations & Transformations › Inverse Functions
What is a Rational Expression?: A rational expression is a ratio of two polynomials, $\dfrac{P(x)}{Q(x)}$ .; Rational Expressions & Functions › Simplifying Rational Expressions & Domain
What is a Sequence?: A sequence is an ordered list of numbers called terms.; Sequences & Series › Sequences: Terms, Explicit & Recursive Rules
What the “arms” do: The end behavior of a graph describes what happens to $f(x)$ as $x\to+\infty$ and $x\to-\infty$ .; Polynomial Functions › End Behavior
What the determinant measures: For $; Linear Systems & Matrices › Determinants and Cramer's Rule
What the two bad rows mean: A row $[\,0\ \ 0\ \ 0\ |\ c\,]$ with $c \ne 0$ reads " $0 = c$ ", which is false.; Linear Systems & Matrices › Reading a Reduction That Did Not Finish
When Does an Infinite Sum Exist?: An infinite geometric series $a_1 + a_1 r + a_1 r^2 + \cdots$ converges (adds up to a finite number) exactly when \[ |r| < 1.; Sequences & Series › Infinite Geometric Series
When is it defined, and what size is the product?: The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ : \[ (m × n) · (n × p) = (m × p).; Linear Systems & Matrices › Matrix Multiplication
When the parabola misses the $x$ -axis: If the discriminant $D<0$ , the graph has no $x$ -intercepts and the two solutions are complex conjugates $p\pm qi$ .; Quadratic Functions & Complex Numbers › Quadratics with Complex Solutions
Where conics show up: Conic Sections › Applications (light)
Where parabolas appear: Quadratics model projectile height, areas, and any quantity with a single maximum or minimum.; Quadratic Functions & Complex Numbers › Modeling with Quadratics
z-Scores: How Many Standard Deviations Out: A z-score rewrites any value $x$ as its distance from the mean, measured in standard deviations: \[ z = $\frac{x - \mu}{\sigma}$ .; Probability & Statistics › The Normal Distribution, the Empirical Rule & z-Scores