https://teddyrookbook.com/math/algebra-2/Recent content in Algebra 2 on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/algebra-2/parent-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/parent-functions/Know the shapes of x, x², x³, |x|, √x, ∛x and 1/x. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/combined-transformations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/combined-transformations/Apply stretches, reflections and shifts in the right order. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/even-odd-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/even-odd-functions/f(−x) = f(x) means symmetric about the y-axis; f(−x) = −f(x) means symmetric about the origin. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/function-operations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/function-operations/Combine functions and find the domain of the result. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/function-composition/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/function-composition/f(g(x)) feeds the output of g into f. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/inverse-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/inverse-functions/An inverse undoes a function; its graph is the reflection over y = x. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/verify-inverses/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/verify-inverses/f(g(x)) = x and g(f(x)) = x confirm an inverse pair. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/restrict-domain-inverse/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/restrict-domain-inverse/Keep one branch of a function so its inverse is a function. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/ramp-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/ramp-functions/ReLU(x) = max(0, x) is a ramp; sums of shifted, scaled ramps make any bent-line graph. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/vertex-formula/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/vertex-formula/The vertex of ax² + bx + c is at x = −b/(2a). Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/quadratic-from-points/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/quadratic-from-points/Build a quadratic from its vertex, its zeros or three points. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/imaginary-unit/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/imaginary-unit/i² = −1, so √−9 = 3i. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/complex-arithmetic/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/complex-arithmetic/Treat i like a variable, then replace i² with −1. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/complex-division/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/complex-division/Multiply by the conjugate to clear i from the denominator. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/complex-solutions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/complex-solutions/A negative discriminant gives two complex conjugate solutions. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/quadratic-inequalities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/quadratic-inequalities/Use the zeros and the graph to find where a quadratic is positive or negative. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/three-variable-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/three-variable-systems/Eliminate one variable twice to reduce to two equations. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/nonlinear-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/nonlinear-systems/Solve systems with circles, parabolas and lines. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/seeing-structure/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/seeing-structure/Treat a chunk like (x + 1) or x² as a single quantity to rewrite an expression. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/end-behavior/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/end-behavior/The leading term decides what happens as x goes to ±∞. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/zeros-multiplicity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/zeros-multiplicity/A zero of even multiplicity touches the axis; odd multiplicity crosses it. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/polynomial-long-division/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/polynomial-long-division/Divide polynomials the way you divide numbers. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/synthetic-division/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/synthetic-division/A shortcut for dividing by x − a. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/remainder-factor-theorems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/remainder-factor-theorems/p(a) is the remainder when dividing by x − a; p(a) = 0 means x − a is a factor. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/sum-difference-cubes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/sum-difference-cubes/a³ ± b³ = (a ± b)(a² ∓ ab + b²). Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/quadratic-form-factoring/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/quadratic-form-factoring/Treat x⁴ − 5x² + 4 as a quadratic in x². Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/rational-root-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/rational-root-theorem/List the possible rational zeros p/q and test them. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/solve-polynomial-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/solve-polynomial-equations/Factor completely and use the zero product property. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/fundamental-theorem-algebra/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/fundamental-theorem-algebra/A degree-n polynomial has n complex zeros, counted with multiplicity. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/polynomial-from-zeros/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/polynomial-from-zeros/Multiply factors for each zero, including complex pairs, and scale to fit a point. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/sketch-polynomials/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/sketch-polynomials/Combine end behavior, zeros and multiplicity to sketch a graph. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/polynomial-identities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/polynomial-identities/Prove identities and use them, such as making Pythagorean triples. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/binomial-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/binomial-theorem/Expand (a + b)ⁿ with coefficients from Pascal's triangle. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/polynomial-inequalities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/polynomial-inequalities/Use a sign chart built from zeros and undefined points. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/simplify-rational/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/simplify-rational/Factor, then cancel common factors, and note excluded values. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/multiply-divide-rational/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/multiply-divide-rational/Factor everything, then multiply as with fractions. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/add-subtract-rational/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/add-subtract-rational/Rewrite over a common denominator. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/complex-fractions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/complex-fractions/Simplify fractions that contain fractions. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/rational-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/rational-equations/Multiply by the common denominator, solve, and reject extraneous solutions. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/rational-models/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/rational-models/Model shared work and average cost with rational equations. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/variation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/variation/y = kx, y = k/x and y = kxz. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/asymptotes-holes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/asymptotes-holes/Find vertical asymptotes, holes and horizontal asymptotes. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/graph-rational-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/graph-rational-functions/Use asymptotes, intercepts and sign to sketch the graph. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/slant-asymptotes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/slant-asymptotes/Divide when the numerator's degree is one more than the denominator's. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/nth-roots/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/nth-roots/ⁿ√a is the number whose nth power is a. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/simplify-radicals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/simplify-radicals/Simplify roots and clear them from denominators. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/radical-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/radical-functions/Graph and transform √x and ∛x. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/radical-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/radical-equations/Isolate the root, raise both sides to a power, and check. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/exponential-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/exponential-graphs/Transform y = bˣ and find its asymptote. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/common-base-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/common-base-equations/Rewrite both sides with the same base and set exponents equal. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/compound-interest/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/compound-interest/A = P(1 + r/n)ⁿᵗ: interest earns interest. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/number-e/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/number-e/Compounding more often approaches A = Peʳᵗ. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/logarithms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/logarithms/log_b(x) is the exponent you put on b to get x. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/log-exp-forms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/log-exp-forms/log_b(y) = x means bˣ = y. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/log-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/log-properties/Logs turn products into sums and powers into multiples. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/change-of-base/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/change-of-base/log_b(x) = ln x / ln b. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/log-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/log-graphs/Graph y = log_b(x) as the reflection of bˣ. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/log-scales/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/log-scales/Each step on a log scale multiplies by the same factor, as in decibels and pH. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/solve-exponential-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/solve-exponential-equations/Take a log of both sides to bring the exponent down. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/solve-log-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/solve-log-equations/Combine logs, rewrite in exponential form, and check the domain. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/exponential-log-models/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/exponential-log-models/Model half-life, doubling time and pH. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/choose-a-model/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/choose-a-model/Match data and situations to the right family of functions. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/recursive-sequences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/recursive-sequences/Define a sequence by its first term and a rule for the next. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/sigma-notation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/sigma-notation/Σ is shorthand for adding the terms of a pattern. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/arithmetic-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/arithmetic-series/Add first and last terms, then multiply by half the number of terms. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/geometric-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/geometric-series/S = a(1 − rⁿ)/(1 − r), and its use for loan payments. Algebra 2 · Math Ladderhttps://teddyrookbook.com/math/algebra-2/infinite-geometric-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/algebra-2/infinite-geometric-series/When |r| < 1 the partial sums approach a/(1 − r). Algebra 2 · Math Ladder