TASC General Mathematics Level 3 cheat sheet

Slope between two points: m = (y2 − y1)/(x2 − x1); a horizontal line has m = 0, a vertical line has undefined slope.

Slope-intercept form y = mx + b: m is the rate of change, b is the y-value where the line crosses the y-axis.

Point-slope form y − y1 = m(x − x1) builds a line from one point and the slope; solve for y to convert to slope-intercept form.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m1 · m2 = −1).

Standard form Ax + By = C: slope is −A/B and y-intercept is C/B when B ≠ 0.

Solve linear systems by substitution (isolate one variable) or elimination (add or subtract scaled equations to cancel a variable).

A system has no solution when the lines are parallel, and infinitely many solutions when the equations describe the same line.

Distribute before combining like terms: a(b + c) = ab + ac; check solutions by substituting back into the original equation.

Linear inequality rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

A function assigns exactly one output to each input; the vertical line test confirms a graph is a function.

Function notation: f(x) means the output for input x, so f(3) replaces every x in the rule with 3.

Domain is the set of valid inputs (exclude values making a denominator zero or a radicand negative); range is the set of outputs.

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

Linear model y = mx + b: m is the constant change per unit and b is the starting value at x = 0.

Exponential model y = a·b^x: a is the initial amount, b > 1 gives growth, 0 < b < 1 gives decay, and b = 1 + r for rate r.

Average rate of change over [a, b] equals (f(b) − f(a))/(b − a), the slope of the secant line.

Inverse of a function: swap x and y, then solve for y; the inverse graph is a reflection over the line y = x.

Quadratic vertex form y = a(x − h)² + k has vertex (h, k); the axis of symmetry is x = h.

Mean = sum of values ÷ count; median = middle value of ordered data; mode = most frequent value.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1, and outliers often fall below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean: small values mean tightly clustered data.

Probability of an event = favorable outcomes ÷ total equally likely outcomes, always between 0 and 1.

Complement rule: P(not A) = 1 − P(A); addition rule: P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); for dependent events multiply by the updated probability of the second event.

Conditional probability: P(A | B) = P(A and B)/P(B) when P(B) > 0.

Expected value = Σ(value × probability); it is the long-run average payoff per trial.

Correlation r ranges from −1 to 1; r near ±1 indicates a strong linear association, and correlation does not imply causation.

Simple interest: I = P·r·t, where P is principal, r is the annual rate as a decimal, and t is time in years.

Compound interest: A = P(1 + r/n)^(nt), with n compounding periods per year; continuous compounding uses A = P·e^(rt).

Compound interest earned equals A − P; more frequent compounding yields a slightly larger total.

Percent change = (new − old)/old × 100; a p% increase multiplies by (1 + p/100), a decrease by (1 − p/100).

Successive percent changes multiply rather than add: a 10% rise then a 10% fall leaves 0.99 of the original amount.

A markup or tax adds a percentage to a base price; a discount subtracts it, and discount is taken before tax in multi-step problems.

Counting rule: if one choice has m options and another has n, the pair has m·n outcomes; permutations count order, combinations do not.

Combinations: nCr = n!/(r!(n − r)!); permutations: nPr = n!/(n − r)!.

A recurrence or sequence term is found by applying the stated rule repeatedly; arithmetic sequences add a constant d, geometric ones multiply by a constant ratio r.

To solve ax + b = cx + d, collect variable terms on one side and constants on the other, then divide by the coefficient.

Factoring: find two numbers whose product is c and sum is b for x² + bx + c; the zero product property then gives the roots.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) solves ax² + bx + c = 0; the discriminant b² − 4ac tells the number of real roots.

Exponent rules: x^a · x^b = x^(a+b), x^a / x^b = x^(a−b), (x^a)^b = x^(ab), and x^0 = 1 for x ≠ 0.

Absolute value equations |x| = k give two cases, x = k and x = −k, when k > 0.

When solving an equation with fractions, multiply every term by the least common denominator to clear it.

Check for extraneous solutions after squaring both sides or multiplying by a variable expression.

Slope of a line on a graph equals rise ÷ run; a steeper line has a larger absolute slope.

x-intercept is found by setting y = 0; y-intercept is found by setting x = 0.

A graph models a situation when the axes carry units and the slope means change in output per unit of input.

Piecewise functions apply different rules on different intervals; evaluate by choosing the rule matching the input.

Composition (f ∘ g)(x) = f(g(x)): evaluate the inner function first, then feed the result into the outer one.

To find where two models are equal, set the expressions equal and solve; the solution is the break-even or intersection input.

Direct variation y = kx passes through the origin; inverse variation y = k/x has a decreasing curve for k > 0.

Extrapolation beyond the data range is less reliable than interpolation within it.

Weighted mean = Σ(value × weight) ÷ Σ(weights); use it when categories have different counts.

Adding a constant to every data value shifts the mean and median by that constant but leaves standard deviation and IQR unchanged.

Multiplying every value by a positive constant multiplies mean, median, standard deviation, and IQR by that constant.

Two-way tables: joint probability uses an inner cell ÷ grand total; marginal probability uses a row or column total ÷ grand total.

Mutually exclusive events cannot occur together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).

Sampling: random selection reduces bias; a larger sample generally gives more reliable estimates of a population.

A line of best fit summarizes trend; residuals (actual − predicted) show how far points fall from the line.

With replacement, draws are independent; without replacement, the total and favorable counts both decrease after each draw.

Monthly payment estimate on a loan uses the amortization formula M = P·i(1 + i)^n / ((1 + i)^n − 1), with i = annual rate ÷ 12 and n = number of months.

Effective annual rate compares compounding choices: EAR = (1 + r/n)^n − 1 for n periods per year.

A budget balances income against fixed and variable expenses; net income = gross income − deductions.

Unit price = total cost ÷ quantity; comparing unit prices identifies the better buy.

Sales tax applies to the taxable subtotal; total cost = subtotal + tax − discount applied in the stated order.

A graph or network problem counts routes or connections; the number of edges meeting at a vertex is its degree.

Greedy scheduling or route choices pick the smallest available option at each step but do not always give the overall best result.

Compound growth doubles an investment in roughly 72 ÷ (annual percent rate) years by the rule of 72.

Translate words to symbols: 'per' signals a rate (slope), 'initial' or 'flat fee' signals the y-intercept.

Break-even point occurs where revenue equals cost; set the two linear expressions equal and solve for quantity.

When a linear model is given as a table, the slope is the constant difference in outputs divided by the constant difference in inputs.

Systems word problems: define variables explicitly, write one equation per relationship, then solve and interpret both values.

Inequality models use ≤ or ≥ for 'at most' or 'at least'; graph the solution region and test a point to check shading.

Direct proportion problems set up ratios equal to each other and cross-multiply to solve for the unknown.

Reading a graph: identify axis labels and units first, then locate intercepts, peaks, and intervals of increase or decrease.

A rate of change read from a graph equals the slope of the segment between two labeled points.

Choose a model by pattern: constant differences suggest linear, constant ratios suggest exponential, and a symmetric peak suggests quadratic.

Domain restrictions come from context: time, distance, and counts cannot be negative, and people or items are whole numbers.

Interpreting the y-intercept means stating the starting value in the units of the problem.

Comparing two models means comparing slopes (rates) and intercepts (starting values) in context.

Choose mean for symmetric data and median when outliers or skew are present.

A box plot displays minimum, Q1, median, Q3, and maximum; the box width equals the IQR.

Experimental probability comes from observed trials; theoretical probability comes from equally likely outcomes.

Simulation: repeat random trials many times and compare the observed proportion to the theoretical probability.

Margin of error shrinks as sample size grows; report an estimate with its uncertainty rather than a single number.

Misleading graphs can result from truncated axes or inconsistent scales; check the scale before comparing heights.

Compare loan offers by total interest paid over the life of the loan, not just the monthly payment.

Savings goal: solve A = P(1 + r/n)^(nt) for t using logarithms, t = ln(A/P) / (n·ln(1 + r/n)).

A wage with overtime: total pay = base rate × regular hours + overtime rate × overtime hours.

Depreciation at a fixed percent per year uses A = P(1 − r)^t, which is exponential decay.

Decision matrices or weighted scoring compare discrete options by assigning weights to criteria and summing weighted scores.

Voting and ranking methods (plurality, runoff, points) can produce different winners from the same ballots.