299 core topics
+ 50 prerequisite topics taught
as needed · approximately 105 hours of instruction
including spaced review
An adaptive diagnostic (up to
40 questions) places the student on the course's knowledge
graph - topics already known are credited, and instruction begins exactly
at the learning frontier. Every topic is taught with a worked-example
lesson and auto-graded practice; a topic is mastered at
75%+ and then maintained through spaced reviews on an
expanding schedule. Mixed checks follow every 6 lessons;
each unit ends with a 12-item quiz, and course-wide assessments appear at
25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather
than skipped, so completion certifies the whole tower, not just the top.
| Function Notation & Evaluation
[E] |
Reading f(x) notation and plugging in inputs. |
| Domain & Range
[E] |
Which inputs a function accepts, and which outputs it can produce. |
| Function Composition
[M] |
Feeding one function's output into another: f(g(x)). |
| Inverse Functions
[M] |
The function that undoes f: f⁻¹(b) is the input that f sends to b. |
| Transformations of Functions
[M] |
How f(x − h) + k slides a graph around the plane. |
| Piecewise Functions
[M] |
Functions defined by different rules on different intervals. |
| Absolute Value Equations
[M] |
|x − a| = b splits into two linear equations. |
| Systems by Elimination
[H] |
Adding or subtracting equations to cancel a variable. |
| Nonlinear Systems
[H] |
Where a line meets a parabola: set the two formulas equal. |
| Complex Numbers
[M] |
The imaginary unit i = √(−1) and numbers of the form a + bi. |
| Operations with Complex Numbers
[M] |
Multiplying complex numbers with FOIL and i² = −1. |
| Quadratics with Complex Roots
[H] |
When the discriminant is negative, the roots come in a conjugate pair a ± bi. |
| Polynomial Division
[H] |
Dividing a polynomial by (x − a) with long or synthetic division. |
| Remainder & Factor Theorems
[M] |
The remainder when p(x) is divided by (x − a) is simply p(a). |
| Zeros of Polynomials
[H] |
Finding all the roots of a cubic by factoring it down. |
| End Behavior of Polynomials
[E] |
Far from the origin, only the leading term matters. |
| Simplifying Rational Expressions
[M] |
Factor top and bottom, then cancel the common factor. |
| Operations on Rational Expressions
[H] |
Multiplying and dividing algebraic fractions. |
| Rational Equations
[H] |
Clearing denominators to solve equations with x below the line. |
| Logarithms
[M] |
log_b(x) asks: to what power must b be raised to get x? |
| Properties of Logarithms
[M] |
Logs turn products into sums, quotients into differences, powers into multiples. |
| Exponential & Log Equations
[M] |
Matching bases and rewriting between exponential and log form. |
| Arithmetic Sequences
[M] |
Sequences that grow by a constant difference each step. |
| Geometric Sequences
[M] |
Sequences that grow by a constant ratio each step. |
| Excluded Values & Domain
[E] |
Factor the denominator to find every input a rational expression forbids. |
| Simplifying Quadratic over Quadratic
[M] |
Factor both trinomials, cancel the shared factor, and read off what's left. |
| Opposite Factors: the −1 Trick
[M] |
a − x and x − a are negatives of each other, so they cancel to −1. |
| Multiplying Rational Expressions
[M] |
Factor every trinomial first, then cancel across the multiplication. |
| Dividing Rational Expressions
[M] |
Flip the second fraction, then factor and cancel like a multiplication. |
| Adding with Unlike Polynomial Denominators
[H] |
Build the common denominator (x + p)(x + q) and combine the numerators. |
| Subtracting Rational Expressions
[H] |
Distribute the minus sign through the entire second numerator. |
| Rational Equations by Cross-Multiplying
[M] |
One fraction equals another: cross-multiply and solve the linear equation. |
| Rational Equations That Turn Quadratic
[H] |
Clearing an x from the denominator leaves a factorable quadratic. |
| Extraneous Solutions
[M] |
A candidate that zeroes an original denominator must be thrown out. |
| Direct Variation
[E] |
y = kx: find the constant from one data point, then predict any other. |
| Inverse Variation
[M] |
y = k/x: the product xy stays constant, so one point predicts the rest. |
| Combined Work-Rate Problems
[H] |
Add the jobs-per-hour rates - 1/a + 1/b = 1/t - and solve for the time. |
| Computing Terms from a Recursive Rule
[E] |
A recursive rule builds each term from the ones before it - step by step. |
| Explicit vs Recursive Definitions
[M] |
The same sequence can be described step-by-step or by a direct formula. |
| From Recursive to Explicit
[M] |
Convert the step rule to a direct formula, then jump straight to term n. |
| Counting Terms of a Sequence
[E] |
Solve a_n = a1 + (n − 1)d for n: divide the total climb by the step size. |
| The Arithmetic Series Formula
[M] |
Sum an arithmetic series as count times the average of the two ends. |
| Solving Inside the Series Formula
[M] |
Run S = n(a1 + an)/2 backwards to recover n, the first, or the last term. |
| Finite Geometric Series
[M] |
Sum a geometric series with Sₙ = a(rⁿ − 1)/(r − 1) - one power, no term list. |
| Evaluating Sigma Notation
[M] |
Read the limits, substitute each index value, and add the results. |
| Writing a Series in Sigma Notation
[M] |
Find the kth-term formula, then set the limits so the ends match. |
| From Sum Formula Back to Terms
[M] |
Subtract consecutive partial sums to recover a single term. |
| Infinite Geometric Series
[M] |
When |r| < 1 the whole endless series adds to exactly a/(1 − r). |
| Repeating Decimals as Fractions
[M] |
A repeating decimal is an infinite geometric series in disguise. |
| Modeling Savings & Loans with Sequences
[H] |
Deposits, payments, and interest are sequences - model them step by step. |
| Exponential ↔ Logarithmic Form
[E] |
Every logarithm is an exponent: b^k = x and log_b(x) = k say the same thing. |
| Evaluating Logarithms Exactly
[E] |
Read log_b(x) as a question: to what power must b be raised to give x? |
| Change of Base
[M] |
log_b(x) = log_c(x)/log_c(b) - rewrite over any convenient common base. |
| Expanding with the Log Laws
[M] |
Products become sums, quotients differences, powers coefficients. |
| Condensing into a Single Logarithm
[M] |
Run the log laws backward: sums into products, coefficients into powers. |
| The Power Law in Detail
[M] |
log_b(x^k) = k·log_b(x) - even when the exponent is a root. |
| Solving Exponential Equations with Logs
[M] |
Take a logarithm of both sides - or match a common base - to free the exponent. |
| Solving Logarithmic Equations
[M] |
Rewrite in exponential form, or combine logs first, then solve for x. |
| Application: the Richter Scale
[M] |
Each whole step up in magnitude is a tenfold jump in amplitude. |
| Application: the pH Scale
[M] |
pH = −log₁₀[H⁺]: a lower pH means an exponentially higher acid concentration. |
| Application: the Decibel Scale
[M] |
Loudness in dB is ten times the base-10 log of the intensity ratio. |
| Application: Doubling Time
[M] |
Repeated doubling is a logarithm base 2: n doublings multiply by 2^n. |
| Synthetic Substitution
[E] |
Horner's method: evaluate a polynomial with only multiplies and adds. |
| Reading the Synthetic-Division Quotient
[M] |
The bottom row of synthetic division is the quotient, then the remainder. |
| Polynomial Long Division
[M] |
Dividing by a quadratic: match leading terms, multiply back, subtract, repeat. |
| The Remainder Theorem
[M] |
The remainder of P(x) ÷ (x − a) is just P(a) - no division required. |
| The Factor Theorem: Finding a Root
[M] |
x = a is a root exactly when (x − a) is a factor, i.e. when P(a) = 0. |
| Testing Whether (x − a) Is a Factor
[M] |
Compute P(a): a zero remainder means (x − a) is a factor. |
| The Rational Root Theorem: Listing Candidates
[M] |
Candidate rational roots are ±(factors of the constant)/(factors of the lead). |
| Finding Integer Roots with the Rational Root Theorem
[M] |
List the candidates, then test them to pin down the actual roots. |
| End Behavior from the Leading Term
[E] |
Degree parity sets whether the ends agree; the lead sign sets the direction. |
| Excluded Values of a Rational Function
[M] |
Every zero of the denominator is barred from the domain. |
| Holes versus Vertical Asymptotes
[M] |
A canceling factor makes a hole; a leftover denominator factor makes an asymptote. |
| Horizontal Asymptotes by Degree
[M] |
Compare top and bottom degrees to read off the horizontal asymptote. |
| Adding Complex Numbers
[E] |
Combine real parts with real parts and imaginary parts with imaginary parts. |
| Subtracting Complex Numbers
[E] |
Distribute the minus sign, then subtract part by part. |
| Multiplying Complex Numbers
[E] |
FOIL the two binomials, then replace i² with −1 and collect parts. |
| Powers of i
[E] |
Powers of i repeat every four steps: i, −1, −i, 1. |
| Complex Conjugates
[E] |
The conjugate of a + bi keeps the real part and flips the sign of i. |
| The Product (a + bi)(a − bi)
[E] |
A number times its conjugate is always the real value a² + b². |
| Dividing Complex Numbers
[M] |
Multiply top and bottom by the denominator's conjugate to clear the i. |
| The Modulus of a Complex Number
[M] |
The modulus |a + bi| = √(a² + b²) is the point's distance from the origin. |
| Adding 2×2 Matrices
[E] |
Add two matrices of the same size entry by matching entry. |
| Scalar Multiplication of a Matrix
[E] |
Multiply a matrix by a number by scaling every entry. |
| Multiplying 2×2 Matrices
[M] |
Each product entry is a row of A dotted with a column of B. |
| The Determinant of a 2×2 Matrix
[M] |
For [[a, b], [c, d]] the determinant is ad − bc. |
| Solving a 2×2 System with Determinants
[M] |
Cramer's rule reads each variable off a ratio of determinants. |
| Coefficients in a Polynomial Product
[H] |
Pick out one coefficient of a product without expanding everything. |
| Sum & Difference of Cubes
[H] |
a³ ± b³ factors into a binomial times an unfactorable trinomial. |
| Factoring by Grouping
[H] |
Pair up four terms, pull a factor from each pair, then share the binomial. |
| Finding a Coefficient from the Remainder
[H] |
Run the Remainder Theorem backwards to pin down an unknown constant. |
| Quartics in Quadratic Form
[H] |
Substitute u = x² to turn a quartic into a familiar quadratic. |
| Sum & Product of Roots (Vieta)
[H] |
Read the sum and product of the roots straight off the coefficients. |
| Complex Fractions
[H] |
Clear a fraction-of-fractions by multiplying through by the inner LCD. |
| Clearing a Denominator
[H] |
Multiply both sides by the denominator, then solve the linear leftovers. |
| Extraneous Roots of Rational Equations
[H] |
Clearing denominators can invent candidates the original equation rejects. |
| Reciprocal-Sum Formulas
[H] |
Solve 1/R = 1/a + 1/b style formulas exactly, not by decimal guessing. |
| Joint & Combined Variation
[H] |
One quantity driven by two others: z = kxy or z = kx/y. |
| Two-Step Radical Equations
[H] |
Isolate the radical first, then square both sides. |
| Extraneous Roots from Squaring
[H] |
Squaring both sides can create a candidate the original equation rejects. |
| Equations with Rational Exponents
[H] |
Undo x^(m/n) by raising both sides to the reciprocal power n/m. |
| Powers of a Complex Number
[H] |
Square or cube a + bi by expanding and folding every i² into -1. |
| Rebuilding a Quadratic from Complex Roots
[H] |
Conjugate roots p ± qi give sum 2p and product p² + q². |
| The Modulus Is Multiplicative
[H] |
|zw| = |z||w|: take moduli first, then multiply plain numbers. |
| Multiplying Square Roots of Negatives
[H] |
Convert to i-form BEFORE multiplying: √(-4)·√(-9) is -6, not 6. |
| Log Equations by Condensing
[H] |
Merge the logs into one, convert to exponential form, then solve. |
| Exponential Equations over a Common Base
[H] |
Rewrite both sides as powers of one base, then equate the exponents. |
| The Natural Logarithm
[H] |
ln is the base-e log: ln(e^k) = k and e^(ln m) = m. |
| Exact Fractional Log Values
[H] |
log_4 8 = 3/2: write base and argument over one prime and divide exponents. |
| How Long to Grow: Doubling & Tripling Time
[H] |
Count how many doublings are needed, then multiply by the doubling time. |
| Half-Life
[H] |
Each half-life cuts the amount in half: after n of them, 1/2^n remains. |
| Exponentials & Logs Undo Each Other
[M] |
log_b(b^k) = k and b^(log_b m) = m, with zero computation. |
| The y-Coordinate of a Hole
[H] |
Cancel the shared factor, then plug the hole's x into what remains. |
| Horizontal Asymptotes, Exactly
[H] |
Equal degrees: y = ratio of leading coefficients. Bottom-heavy: y = 0. |
| Intercepts of Rational Functions
[H] |
x-intercepts come from the numerator; the y-intercept is f(0). |
| Slant Asymptotes by Division
[H] |
Numerator one degree heavier: divide, keep the linear quotient. |
| Where a Line Meets a Circle
[H] |
Substitute the line into the circle and solve the resulting quadratic. |
| Systems in Three Variables
[H] |
Combine three equations to peel off one variable at a time. |
| Sum & Product Systems
[H] |
Numbers with sum s and product p are the roots of t² - st + p = 0. |
| Finite Geometric Sums in Context
[H] |
Total a + ar + ... + ar^(n-1) = a(r^n - 1)/(r - 1). |
| Sigma Notation: Geometric Sums
[H] |
Read the limits carefully, then apply the geometric sum formula. |
| Sigma Notation: Linear Sums
[H] |
Split Σ(ak + b) into a·Σk + b·n and use Σk = n(n+1)/2. |
| Geometric Means
[H] |
The middle of a geometric triple is the square root of the outer product. |
| Composition as a Formula
[H] |
Build f(g(x)) as a new rule by substituting g's whole formula into f. |
| The Inverse as a Formula
[H] |
Swap x and y, then solve for y: the steps of f, undone in reverse order. |
| Reading an Inverse from a Table
[M] |
f⁻¹(b) = a means the table row where f outputs b: read it backwards. |
| Tracking a Point through Transformations
[H] |
y = a·f(x - h) + k sends (p, q) to (p + h, aq + k). |
| Recognizing a Transformed Parent Function
[H] |
Inside the function moves left-right (backwards); outside moves up-down. |
| Even & Odd Functions
[H] |
Even: f(-x) = f(x), mirror symmetry. Odd: f(-x) = -f(x). |
| Degrees in a Division Statement
[H] |
Quotient degree is dividend minus divisor; the remainder stays below the divisor. |
| Dividing When Powers Are Missing
[H] |
Insert a zero coefficient for every absent power before dividing. |
| Dividing by (ax - b)
[H] |
Synthetic division uses the root b/a, then the quotient row is divided by a. |
| Rebuilding the Dividend
[H] |
Multiply divisor by quotient and add the remainder to recover p(x). |
| The Remainder Left by a Quadratic Divisor
[H] |
Dividing by (x - a)(x - b) leaves mx + n, fixed by the values p(a) and p(b). |
| The Remainder Theorem at a Fractional Root
[H] |
The remainder on division by (ax - b) is p(b/a), which is often a fraction. |
| A Coefficient Forced by a Required Factor
[H] |
Set p at the divisor's root equal to zero and solve for the unknown coefficient. |
| Two Coefficients from Two Conditions
[H] |
Each stated factor or value gives one equation in the unknown coefficients. |
| Finishing a Factorization from One Zero
[H] |
Divide out the known factor, then factor the smaller quotient completely. |
| Ruling a Value Out with the Rational Root Theorem
[H] |
A rational zero p/q needs p dividing the constant and q dividing the leading coefficient. |
| Finding a Fractional Zero
[H] |
When the leading coefficient is not 1, the rational zero can be a fraction p/q. |
| Conjugate Pairs Forced by Real Coefficients
[H] |
Nonreal zeros of a real polynomial arrive in conjugate pairs, so their count is even. |
| Building a Cubic from a Conjugate Pair
[H] |
A conjugate pair contributes the real quadratic factor x² - 2px + (p² + q²). |
| The Real Zero a Conjugate Pair Leaves Behind
[H] |
Divide out the quadratic from the conjugate pair, or use the sum of the zeros. |
| A Quartic from Two Conjugate Pairs
[H] |
Two conjugate pairs give two real quadratic factors whose product is the quartic. |
| Writing a Polynomial from Its Zeros
[H] |
Each zero r contributes a factor (x - r); multiply them out for standard form. |
| The Leading Coefficient Fixed by a Point
[H] |
Zeros determine the factors; one extra point determines the leading coefficient. |
| Radical Conjugate Zeros
[H] |
Rational coefficients force a + √k to be paired with a - √k. |
| Degree and Multiplicity
[H] |
The multiplicities of all zeros add up to the degree. |
| The Least Degree a Graph Can Have
[H] |
Crossings need odd multiplicity, touches need even, and nonreal zeros come in pairs. |
| The Leading Term of a Factored Form
[H] |
Multiply the leading term of each factor, raised to that factor's power. |
| Crossing, Touching, and Flattening
[H] |
Odd multiplicity crosses the axis, even multiplicity touches and turns back. |
| Counting the Intervals Where a Polynomial Is Positive
[H] |
Sign can only change at a zero of odd multiplicity, so alternation can skip. |
| Adding Three Rational Expressions
[H] |
Build one common denominator for all three terms, then combine numerators. |
| Multiply-Divide Chains
[H] |
Invert every divisor, factor everything, cancel, and only then substitute. |
| Least Common Denominators with Repeated Factors
[H] |
The LCD takes each distinct factor to its highest power anywhere. |
| Canceling with Sums and Differences of Cubes
[H] |
x³ - a³ = (x - a)(x² + ax + a²), which cancels against a linear factor. |
| Complex Fractions with Binomial Denominators
[H] |
Multiply the whole complex fraction, top and bottom, by the inner LCD. |
| Continued Fractions
[H] |
Simplify a nested fraction from the innermost level outward. |
| Rational Equations That Clear to a Quadratic
[H] |
Multiply by the LCD, collect everything on one side, and factor the quadratic. |
| Which Candidates Survive the Check
[H] |
Any candidate that zeroes an original denominator must be discarded. |
| Motion Problems with Wind and Current
[H] |
Time equals distance over rate, so wind and current problems are rational equations. |
| Behavior on Each Side of a Vertical Asymptote
[H] |
A test value just past the asymptote gives the sign, hence the direction. |
| Where a Graph Crosses Its Horizontal Asymptote
[H] |
Set f(x) equal to the asymptote's y-value and solve the resulting equation. |
| Repeated Factors: Hole or Asymptote?
[H] |
Compare the powers of the shared factor: leftovers in the denominator still make an asymptote. |
| Classifying the End-Behavior Asymptote
[H] |
Degree gap 0 gives a horizontal line, gap 1 a slant line, more than 1 neither. |
| Building a Rational Function from Its Features
[H] |
Zeros go in the numerator, asymptotes in the denominator, and the ratio sets the horizontal line. |
| Solving a Polynomial Inequality
[H] |
Mark the zeros, test one point per interval, and collect the intervals that qualify. |
| Inequalities with Repeated Factors
[H] |
A factor of even multiplicity touches zero without changing the sign. |
| Solving a Rational Inequality
[H] |
Sign changes happen at zeros of the numerator AND of the denominator. |
| Counting the Integers That Satisfy an Inequality
[H] |
Solve the inequality first, then count the integers inside the solution set. |
| The Extreme Integer Solution
[H] |
Factor first, build the sign chart, then read off the extreme integer allowed. |
| Radical Form and Rational Exponent Form
[H] |
The n-th root of x to the m equals x raised to the power m/n. |
| Evaluating Powers with Rational Exponents
[H] |
Take the root named by the denominator first, then raise to the numerator. |
| Combining Fractional Exponents
[H] |
Multiply powers by adding fractional exponents, divide by subtracting. |
| Roots of Variable Powers
[H] |
Divide each exponent by the index; pull out the largest perfect power. |
| Domains of Even and Odd Roots
[H] |
Even roots need a nonnegative radicand; odd roots accept every real number. |
| Rational Exponent Models
[H] |
Power models y = kx^(m/n) are evaluated and inverted with reciprocal powers. |
| Simplifying Larger Square Roots
[H] |
Split the radicand into the largest perfect square times the rest. |
| Simplifying Cube Roots
[H] |
Pull out perfect CUBE factors: a factor must appear three times to escape. |
| Adding Radicals After Simplifying
[H] |
Radicals combine only after each one is reduced to the same radicand. |
| Multiplying Radical Binomials
[H] |
FOIL as usual, then replace every √m·√m with m and collect like parts. |
| Rationalizing a Monomial Denominator
[H] |
Multiply top and bottom by whatever completes the root in the denominator. |
| Rationalizing with a Conjugate
[H] |
Multiply top and bottom by the conjugate to make the denominator rational. |
| Radical Equations with Two Radicals
[H] |
Two radicals equal to each other square away in one step; a sum needs two. |
| Checking Candidates of a Radical Equation
[H] |
Squaring can add candidates, so test each one in the original equation. |
| Cube Root Equations
[H] |
Cubing both sides is reversible, so a cube root equation has no extraneous roots. |
| Solving a Radical Formula
[H] |
Isolate the radical in a science formula, then square to free the variable. |
| Radical Equations in Quadratic Form
[H] |
Substitute u for the radical: x - k√x + m = 0 is a quadratic in √x. |
| Verifying an Inverse by Composition
[H] |
A pair are inverses exactly when both compositions return the input x. |
| The Inverse of a Radical Function
[H] |
Swap x and y, then square to undo the square root. |
| The Inverse of a Cubic Function
[H] |
Undo a cube with a cube root, working from the outside operation inward. |
| Restricting a Domain to Invert
[H] |
A parabola inverts only after its domain is cut to one side of the vertex. |
| Where a Function Meets Its Inverse
[H] |
For an increasing function, f and its inverse meet where f(x) = x. |
| The Inverse of an Exponential Function
[H] |
The inverse of b^x is log_b x: it reports the exponent that produced a value. |
| The Domain of a Logarithmic Function
[H] |
A logarithm accepts only positive arguments, and its graph rises from that boundary. |
| Evaluating a Transformed Logarithmic Function
[H] |
Undo the shift and stretch around the logarithm, one operation at a time. |
| Logs of Composite Numbers from Known Values
[H] |
Factor the argument into known pieces, then apply the product, quotient and power rules. |
| Log Properties: Which Rules Are Real
[H] |
Logs convert products to sums, but a log of a SUM cannot be broken apart. |
| Expanding a Logarithm: Reading the Coefficients
[H] |
Every factor becomes a term whose coefficient is its exponent, negative below the bar. |
| Condensing Logarithms of Expressions
[H] |
Coefficients become exponents; sums become products and differences quotients. |
| Natural Log Properties
[H] |
The log laws apply to ln, with the extra simplification ln e = 1. |
| Writing a Log with the Change of Base Formula
[H] |
log_b N equals log N over log b in any single convenient base. |
| Chaining Logarithms
[H] |
log_a b times log_b c collapses to log_a c: the middle base cancels. |
| Bounding a Logarithm Between Integers
[H] |
Trap the argument between two powers of the base to bracket the logarithm. |
| Exponential Equations with a Quadratic Exponent
[H] |
Equal powers of one base force equal exponents, even when an exponent is quadratic. |
| Exponential Equations in Quadratic Form
[H] |
Since b^(2x) = (b^x)², the substitution u = b^x turns the equation into a quadratic. |
| Factoring Out a Common Exponential
[H] |
b^(x+k) is b^x times b^k, so a common power factors out of a sum of exponentials. |
| Exponential Equations with Unlike Bases
[H] |
Take a logarithm of both sides so the power law brings the exponent down. |
| Logarithmic Equations with a Log on Each Side
[H] |
One-to-one logs: if log_b A = log_b B then A = B, provided both stay positive. |
| Domain Checks in Logarithmic Equations
[H] |
Condensing can produce candidates that make a logarithm undefined, and sometimes none survive. |
| Equations Quadratic in a Logarithm
[H] |
Substitute u for the logarithm, solve the quadratic, then convert each u back. |
| Compound Interest with Several Periods a Year
[H] |
A = P(1 + r/n)^(nt): divide the rate by n and multiply the years by n. |
| Continuous Compounding
[H] |
A = Pe^(rt): the growth factor over t years is e^(rt), so rt fixes the multiple. |
| Population Growth Models
[H] |
A constant percent increase means multiplying by the same factor 1 + r every year. |
| Finding a Growth Factor from Two Data Points
[H] |
The annual factor is the k-th root of the ratio of the two measurements. |
| Writing a Half-Life Model
[H] |
Half-life h gives A = A0(1/2)^(t/h): the exponent counts how many half-lives have passed. |
| Newton's Law of Cooling: Temperature
[H] |
Only the difference from the surroundings decays; the surrounding temperature is added back. |
| Newton's Law of Cooling: Time
[H] |
Count how many times the temperature difference must halve, then multiply by the halving time. |
| Arithmetic Totals in Context
[H] |
Seating, stacking and production totals are arithmetic series in words. |
| Summing a Block of an Arithmetic Series
[H] |
The sum of terms m through n is the count of terms times the average of the two ends. |
| Sigma Notation with a Shifted Lower Limit
[H] |
A lower limit other than 1 changes the term count, not the term rule. |
| Re-Indexing a Sum
[H] |
Shifting the index shifts the limits one way and the term rule the other way. |
| Summing a Block of a Geometric Series
[H] |
Add terms m through n of a geometric sequence as S(n) minus S(m-1). |
| How Many Terms Reach a Geometric Total
[H] |
Run S = a(rⁿ - 1)/(r - 1) backwards to recover the number of terms. |
| Recovering the Ratio from a Geometric Sum
[H] |
Divide the sum by the first term, then solve the resulting polynomial in r. |
| Does the Infinite Series Converge?
[H] |
An infinite geometric series has a finite sum exactly when the ratio satisfies |r| < 1. |
| Infinite Geometric Series in Sigma Notation
[H] |
Read the first term straight off the lower limit, then apply a/(1 - r). |
| Total Path of a Bouncing Ball
[H] |
Rebound heights form an infinite geometric series; the path counts each rise twice. |
| The Long-Run Level of a Recursive Model
[H] |
When a fraction r is retained and a fixed amount c is added, the level settles at c/(1 - r). |
| A Sequence Rebuilt from Two of Its Terms
[H] |
Two terms fix the step: divide the change by the gap in index positions. |
| A Circle Recovered from General Form
[H] |
Complete the square in x and in y to turn Ax² + Ay² + Dx + Ey + F = 0 into standard form. |
| Focus and Directrix of a Parabola
[H] |
In (x - h)² = 4p(y - k) the focus sits p units from the vertex and the directrix p units the other way. |
| Building a Parabola from Focus and Directrix
[H] |
The vertex is halfway between focus and directrix, and p is the distance to either one. |
| Horizontal Parabolas
[H] |
When y is squared the parabola opens left or right and the roles of x and y swap. |
| Parabolic Dishes and Arches
[H] |
Put the vertex at the origin, substitute the rim point, and solve for the focal distance. |
| Center, Vertices and Axes of an Ellipse
[H] |
The larger denominator names the major axis; its square root is the semi-axis length. |
| Foci of an Ellipse
[H] |
For an ellipse the focal distance satisfies c² = a² - b², with a the semi-major axis. |
| Writing the Equation of an Ellipse
[H] |
Vertices give a, foci give c, and b² = a² - c² fills in the other denominator. |
| Elliptical Arches and Whispering Galleries
[H] |
A semi-elliptical arch of width 2a and height b satisfies x²/a² + y²/b² = 1 for y at least 0. |
| Vertices and Transverse Axis of a Hyperbola
[H] |
The variable with the POSITIVE term names the transverse axis, whatever the denominators are. |
| Foci of a Hyperbola
[H] |
For a hyperbola the focal distance satisfies c² = a² + b², so the foci lie beyond the vertices. |
| Asymptotes of a Hyperbola
[H] |
The asymptotes pass through the center with slopes given by the ratio of the semi-axes. |
| Writing the Equation of a Hyperbola
[H] |
Vertices give a, foci or asymptote slope give b, and the positive term follows the transverse axis. |
| Identifying a Conic from Its Equation
[H] |
Compare the two squared terms: their count, their signs and whether their coefficients match. |
| Where a Line Meets an Ellipse
[H] |
Substitute the line into the ellipse and solve the quadratic that results. |
| Where a Line Meets a Hyperbola
[H] |
Substituting a line into a hyperbola gives a quadratic, unless the line matches an asymptote slope. |
| Two Conics at Once
[H] |
Treat x² and y² as the two unknowns, eliminate one, then take square roots. |
| How Many Intersections?
[H] |
Substitute, collect a quadratic, and let the discriminant count the real intersections. |
| Arrangements with Repeated Letters
[H] |
Divide n! by a factorial for each repeated letter, since swapping identical letters changes nothing. |
| Circular Arrangements
[H] |
Seat one person to fix the rotation, then arrange the remaining n - 1 people in a line. |
| Committees with Restrictions
[H] |
Fix the forced members first, multiply independent choices, and use the complement for at least one. |
| Counting by Complement
[H] |
Count everything, then subtract the arrangements that fail the condition. |
| Pascal's Triangle and Binomial Coefficients
[H] |
Entry k of row n of Pascal's triangle is exactly the combination C(n, k). |
| A Specified Term of a Binomial Expansion
[H] |
The rth term of (a + b)^n is C(n, r - 1)·a^(n-r+1)·b^(r-1). |
| The Coefficient of a Chosen Power
[H] |
Choose the index k that makes the powers of x combine to the exponent you want. |
| Conditional Probability from a Two-Way Table
[H] |
Conditioning on an event replaces the grand total by that event's own total. |
| Testing Independence in a Two-Way Table
[H] |
Two events are independent exactly when conditioning on one leaves the other's probability unchanged. |
| The Conditional Probability Formula
[H] |
P(A | B) = P(A and B)/P(B), and the same identity rearranges to P(A and B) = P(A | B)·P(B). |
| The General Addition Rule
[H] |
P(A or B) = P(A) + P(B) - P(A and B), because the overlap is otherwise counted twice. |
| Reversing a Conditional with a Tree
[H] |
P(B | A) is the branch that produced A divided by every branch that produces A. |
| The Binomial Probability Model
[H] |
P(exactly k successes in n independent trials) = C(n, k)p^k(1 - p)^(n-k). |
| Areas from a Standard Normal Table
[H] |
Convert the boundary to a z-score, read the area to its left, then adjust for the region asked about. |
| From a Percentile Back to a Raw Score
[H] |
Look up the area inside the table to recover z, then undo the standardization with x = mu + z*sigma. |
| Recovering the Mean or the Standard Deviation
[H] |
Solve z = (x - mu)/sigma for whichever of the three quantities is unknown. |
| Equivalent Scores Across Two Distributions
[H] |
Two scores represent the same standing exactly when they share a z-score. |