OST Algebra I cheat sheet

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n·b^n, a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Rational exponents: a^(m/n) = the n-th root of a^m = (the n-th root of a)^m; the root is taken first when a is negative and n is even, which is undefined in the reals.

Radical simplification: √(ab) = √a·√b and √(a/b) = √a/√b; rationalize a denominator by multiplying by the conjugate, e.g. 1/(√3+1) → (√3−1)/2.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply by adding exponents, divide by subtracting, and raise to a power by multiplying the exponent by that power.

Classify numbers: natural ⊂ whole ⊂ integers ⊂ rational ⊂ real; irrational numbers (√2, π) have non-terminating, non-repeating decimals and are not expressible as p/q.

Units and rates: convert with dimensional analysis so unwanted units cancel; a rate such as 60 km/h equals 60,000 m / 3,600 s ≈ 16.7 m/s.

Percent change = (new − old)/old × 100%; successive percent changes multiply, so +20% then −20% yields 0.96 of the original, not 100%.

Rational expressions: factor numerator and denominator, cancel common factors, and state excluded values where any denominator equals zero.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have slopes whose product is −1 (negative reciprocals).

Linear forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), standard Ax + By = C; a horizontal line has slope 0, a vertical line has undefined slope.

Solve systems by substitution or elimination; a system has one solution if slopes differ, none if slopes are equal but intercepts differ, and infinitely many if the equations are equivalent.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and no real roots if negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form y = a(x − h)² + k shows the vertex (h, k) and the axis of symmetry x = h.

Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)²; for x² + bx + c find factors of c summing to b.

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where a is the initial value, r the rate as a decimal, and t the number of periods.

Polynomial arithmetic: multiply with the distributive property, divide by factoring or long division, and use the remainder theorem - dividing by (x − k) leaves remainder equal to the polynomial evaluated at k.

Literal equations: isolate a target variable by applying inverse operations to both sides in reverse order of operations; e.g. solve A = P(1 + rt) for t to get t = (A/P − 1)/r.

Function definition: each input in the domain maps to exactly one output; the vertical line test confirms a graph is a function, and the horizontal line test confirms one-to-one (invertible) behavior.

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, and f⁻¹ exists only when f is one-to-one, with f⁻¹(f(x)) = x.

Domain restrictions: exclude inputs making a denominator zero or a radicand negative; for an even root, solve radicand ≥ 0; for a rational expression, exclude zeros of the denominator.

Transformations of f(x): f(x) + k shifts up by k, f(x − h) shifts right by h, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, and a·f(x) stretches vertically by |a|.

Average rate of change over [a, b] = (f(b) − f(a))/(b − a); this is the slope of the secant line joining the two points on the graph.

Parent function shapes: linear y = x; quadratic y = x² (parabola); absolute value y = |x| (V-shape); exponential y = b^x (asymptote y = 0); square root y = √x (domain x ≥ 0).

Exponential vs linear: linear functions change by a constant difference per step, while exponential functions change by a constant ratio per step; growth if base > 1, decay if 0 < base < 1.

Arithmetic sequences follow aₙ = a₁ + (n − 1)d; geometric sequences follow aₙ = a₁·r^(n−1); each is a function on the natural numbers with constant difference or ratio.

Measures of center: mean = sum/n; median = middle value of ordered data (average of two middle values if n is even); mode = most frequent value; the mean is pulled toward outliers more than the median.

Interquartile range IQR = Q3 − Q1; an outlier is commonly a value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean: larger spread yields a larger standard deviation; adding a constant to every value leaves it unchanged.

Two-way tables: joint probability P(A and B) = count in the cell / total; conditional probability P(A | B) = count in cell / row or column total for B.

Probability rules: 0 ≤ P(E) ≤ 1; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); independent events satisfy P(A and B) = P(A)·P(B).

Line of best fit: the correlation coefficient r ranges from −1 to 1; |r| near 1 indicates a strong linear association, and r near 0 indicates little linear association.

Residual = observed − predicted; a residual plot with no pattern supports a linear model, while a curved pattern suggests a nonlinear relationship.

Sampling: random sampling reduces bias; a larger sample size generally reduces variability of a sample statistic; a sample survey supports inference only about the population sampled.

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n·b^n, a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Rational exponents: a^(m/n) = the n-th root of a^m = (the n-th root of a)^m; the root is taken first when a is negative and n is even, which is undefined in the reals.

Radical simplification: √(ab) = √a·√b and √(a/b) = √a/√b; rationalize a denominator by multiplying by the conjugate, e.g. 1/(√3+1) → (√3−1)/2.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply by adding exponents, divide by subtracting, and raise to a power by multiplying the exponent by that power.

Classify numbers: natural ⊂ whole ⊂ integers ⊂ rational ⊂ real; irrational numbers (√2, π) have non-terminating, non-repeating decimals and are not expressible as p/q.

Units and rates: convert with dimensional analysis so unwanted units cancel; a rate such as 60 km/h equals 60,000 m / 3,600 s ≈ 16.7 m/s.

Percent change = (new − old)/old × 100%; successive percent changes multiply, so +20% then −20% yields 0.96 of the original, not 100%.

Rational expressions: factor numerator and denominator, cancel common factors, and state excluded values where any denominator equals zero.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have slopes whose product is −1 (negative reciprocals).

Linear forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), standard Ax + By = C; a horizontal line has slope 0, a vertical line has undefined slope.

Solve systems by substitution or elimination; a system has one solution if slopes differ, none if slopes are equal but intercepts differ, and infinitely many if the equations are equivalent.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and no real roots if negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form y = a(x − h)² + k shows the vertex (h, k) and the axis of symmetry x = h.

Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)²; for x² + bx + c find factors of c summing to b.

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where a is the initial value, r the rate as a decimal, and t the number of periods.

Polynomial arithmetic: multiply with the distributive property, divide by factoring or long division, and use the remainder theorem - dividing by (x − k) leaves remainder equal to the polynomial evaluated at k.

Literal equations: isolate a target variable by applying inverse operations to both sides in reverse order of operations; e.g. solve A = P(1 + rt) for t to get t = (A/P − 1)/r.

Function definition: each input in the domain maps to exactly one output; the vertical line test confirms a graph is a function, and the horizontal line test confirms one-to-one (invertible) behavior.

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, and f⁻¹ exists only when f is one-to-one, with f⁻¹(f(x)) = x.

Domain restrictions: exclude inputs making a denominator zero or a radicand negative; for an even root, solve radicand ≥ 0; for a rational expression, exclude zeros of the denominator.

Transformations of f(x): f(x) + k shifts up by k, f(x − h) shifts right by h, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, and a·f(x) stretches vertically by |a|.

Average rate of change over [a, b] = (f(b) − f(a))/(b − a); this is the slope of the secant line joining the two points on the graph.

Parent function shapes: linear y = x; quadratic y = x² (parabola); absolute value y = |x| (V-shape); exponential y = b^x (asymptote y = 0); square root y = √x (domain x ≥ 0).

Exponential vs linear: linear functions change by a constant difference per step, while exponential functions change by a constant ratio per step; growth if base > 1, decay if 0 < base < 1.

Arithmetic sequences follow aₙ = a₁ + (n − 1)d; geometric sequences follow aₙ = a₁·r^(n−1); each is a function on the natural numbers with constant difference or ratio.

Measures of center: mean = sum/n; median = middle value of ordered data (average of two middle values if n is even); mode = most frequent value; the mean is pulled toward outliers more than the median.

Interquartile range IQR = Q3 − Q1; an outlier is commonly a value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean: larger spread yields a larger standard deviation; adding a constant to every value leaves it unchanged.

Two-way tables: joint probability P(A and B) = count in the cell / total; conditional probability P(A | B) = count in cell / row or column total for B.

Probability rules: 0 ≤ P(E) ≤ 1; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); independent events satisfy P(A and B) = P(A)·P(B).

Line of best fit: the correlation coefficient r ranges from −1 to 1; |r| near 1 indicates a strong linear association, and r near 0 indicates little linear association.

Residual = observed − predicted; a residual plot with no pattern supports a linear model, while a curved pattern suggests a nonlinear relationship.

Sampling: random sampling reduces bias; a larger sample size generally reduces variability of a sample statistic; a sample survey supports inference only about the population sampled.

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n·b^n, a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Rational exponents: a^(m/n) = the n-th root of a^m = (the n-th root of a)^m; the root is taken first when a is negative and n is even, which is undefined in the reals.

Radical simplification: √(ab) = √a·√b and √(a/b) = √a/√b; rationalize a denominator by multiplying by the conjugate, e.g. 1/(√3+1) → (√3−1)/2.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply by adding exponents, divide by subtracting, and raise to a power by multiplying the exponent by that power.

Classify numbers: natural ⊂ whole ⊂ integers ⊂ rational ⊂ real; irrational numbers (√2, π) have non-terminating, non-repeating decimals and are not expressible as p/q.

Units and rates: convert with dimensional analysis so unwanted units cancel; a rate such as 60 km/h equals 60,000 m / 3,600 s ≈ 16.7 m/s.

Percent change = (new − old)/old × 100%; successive percent changes multiply, so +20% then −20% yields 0.96 of the original, not 100%.

Rational expressions: factor numerator and denominator, cancel common factors, and state excluded values where any denominator equals zero.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have slopes whose product is −1 (negative reciprocals).

Linear forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), standard Ax + By = C; a horizontal line has slope 0, a vertical line has undefined slope.

Solve systems by substitution or elimination; a system has one solution if slopes differ, none if slopes are equal but intercepts differ, and infinitely many if the equations are equivalent.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and no real roots if negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form y = a(x − h)² + k shows the vertex (h, k) and the axis of symmetry x = h.

Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)²; for x² + bx + c find factors of c summing to b.

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where a is the initial value, r the rate as a decimal, and t the number of periods.

Polynomial arithmetic: multiply with the distributive property, divide by factoring or long division, and use the remainder theorem - dividing by (x − k) leaves remainder equal to the polynomial evaluated at k.

Literal equations: isolate a target variable by applying inverse operations to both sides in reverse order of operations; e.g. solve A = P(1 + rt) for t to get t = (A/P − 1)/r.

Function definition: each input in the domain maps to exactly one output; the vertical line test confirms a graph is a function, and the horizontal line test confirms one-to-one (invertible) behavior.

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, and f⁻¹ exists only when f is one-to-one, with f⁻¹(f(x)) = x.

Domain restrictions: exclude inputs making a denominator zero or a radicand negative; for an even root, solve radicand ≥ 0; for a rational expression, exclude zeros of the denominator.

Transformations of f(x): f(x) + k shifts up by k, f(x − h) shifts right by h, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, and a·f(x) stretches vertically by |a|.

Average rate of change over [a, b] = (f(b) − f(a))/(b − a); this is the slope of the secant line joining the two points on the graph.

Parent function shapes: linear y = x; quadratic y = x² (parabola); absolute value y = |x| (V-shape); exponential y = b^x (asymptote y = 0); square root y = √x (domain x ≥ 0).

Exponential vs linear: linear functions change by a constant difference per step, while exponential functions change by a constant ratio per step; growth if base > 1, decay if 0 < base < 1.

Arithmetic sequences follow aₙ = a₁ + (n − 1)d; geometric sequences follow aₙ = a₁·r^(n−1); each is a function on the natural numbers with constant difference or ratio.

Measures of center: mean = sum/n; median = middle value of ordered data (average of two middle values if n is even); mode = most frequent value; the mean is pulled toward outliers more than the median.

Interquartile range IQR = Q3 − Q1; an outlier is commonly a value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean: larger spread yields a larger standard deviation; adding a constant to every value leaves it unchanged.

Two-way tables: joint probability P(A and B) = count in the cell / total; conditional probability P(A | B) = count in cell / row or column total for B.

Probability rules: 0 ≤ P(E) ≤ 1; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); independent events satisfy P(A and B) = P(A)·P(B).

Line of best fit: the correlation coefficient r ranges from −1 to 1; |r| near 1 indicates a strong linear association, and r near 0 indicates little linear association.

Residual = observed − predicted; a residual plot with no pattern supports a linear model, while a curved pattern suggests a nonlinear relationship.

Sampling: random sampling reduces bias; a larger sample size generally reduces variability of a sample statistic; a sample survey supports inference only about the population sampled.