TASC Mathematics Methods – Foundation Level 3 cheat sheet

Slope of a line through points (x1,y1) and (x2,y2): m = (y2 - y1)/(x2 - x1); a vertical line has undefined slope.

Slope-intercept form y = mx + b: m is the rate of change and b is the y-intercept; point-slope form is y - y1 = m(x - x1).

Quadratic formula: for ax^2 + bx + c = 0 with a not 0, x = [-b ± sqrt(b^2 - 4ac)]/(2a).

Discriminant D = b^2 - 4ac: D > 0 gives two real roots, D = 0 gives one repeated root, D < 0 gives no real roots.

Vertex of y = ax^2 + bx + c lies at x = -b/(2a); the axis of symmetry is that same vertical line.

Exponent rules: a^m * a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn); a^0 = 1 for a not 0; a^(-n) = 1/a^n.

A relation is a function only if each input has exactly one output; the vertical line test fails a graph that is not a function.

For f(x), the domain excludes values making a denominator zero or making an even root's radicand negative.

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

Derivative as a limit: f'(x) = lim(h->0) [f(x+h) - f(x)]/h; it gives the instantaneous rate of change.

Power rule: d/dx of x^n is n*x^(n-1) for any real n; the derivative of a constant is 0.

Sum and constant rules: d/dx [f + g] = f' + g'; d/dx [cf] = cf'.

At a local maximum or minimum of a differentiable function, f'(x) = 0; f' > 0 means increasing, f' < 0 means decreasing.

Second derivative f'' > 0 indicates concave up; f'' < 0 indicates concave down.

For position s(t), velocity is s'(t) and acceleration is s''(t); velocity zero marks a possible turning point.

Definite integral of a rate over an interval gives total change: integral from a to b of f'(x) dx = f(b) - f(a).

Probability of an event satisfies 0 <= P(E) <= 1; P(not E) = 1 - P(E).

Addition rule: P(A or B) = P(A) + P(B) - P(A and B); for mutually exclusive events the last term is 0.

Multiplication rule: P(A and B) = P(A)*P(B|A); if A and B are independent, P(A and B) = P(A)*P(B).

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

Mean of a data set is the sum of values divided by the count; median is the middle value of ordered data; mode is most frequent.

Standard deviation measures typical distance from the mean; a larger spread gives a larger standard deviation.

In a normal distribution about 68% of values lie within 1 standard deviation of the mean and about 95% within 2.

Outliers can pull the mean toward them while the median stays resistant to extreme values.

Define variables with units before writing equations; every term in an equation must share the same units.

Linear model y = mx + b suits constant-rate change; exponential model y = a*b^x suits repeated percent change with b = 1 + r.

Exponential growth uses b > 1; exponential decay uses 0 < b < 1; the multiplier per period is 1 plus the decimal rate.

Compound interest: A = P(1 + r/n)^(nt), where n is compounding periods per year and t is years.

Check a model by substituting known data points and verifying predicted values against the original situation.

Identify constraints and reasonable domain: lengths, times, and counts are typically nonnegative.

Unit conversion factors equal 1, so multiplying by them changes units without changing the value.

A model's limitations include the range of data used, ignored factors, and assumptions such as constant rate.

System of two linear equations: solve by substitution or elimination; parallel lines (equal slopes, different intercepts) give no solution.

Factoring: find two numbers whose product is c and sum is b to factor x^2 + bx + c.

Absolute value equation |x| = k has solutions x = k and x = -k when k > 0, and no solution when k < 0.

Function composition: (f o g)(x) = f(g(x)); the inner function is evaluated first.

Inverse function f^(-1) reverses f: if f(a) = b then f^(-1)(b) = a.

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

Product rule: d/dx [f*g] = f'g + fg'; quotient rule: d/dx [f/g] = (f'g - fg')/g^2.

Chain rule: d/dx f(g(x)) = f'(g(x))*g'(x); differentiate the outside then multiply by the inside derivative.

Tangent line at x = a has slope f'(a) and equation y - f(a) = f'(a)(x - a).

A function is continuous at a point when the limit exists, the value exists, and the two are equal.

Critical points occur where f'(x) = 0 or f' is undefined; test intervals or f'' to classify them.

Instantaneous rate of change of f at x = a is f'(a), the limit of average rates over shrinking intervals.

Mean of a discrete random variable: E(X) = sum of x*P(x) over all outcomes.

Expected value is a long-run average, not a guaranteed outcome of any single trial.

Counting: if a task has m ways for one step and n for the next, the total is m*n; permutations order items, combinations do not.

Combinations formula: C(n,r) = n!/[r!(n-r)!]; permutations formula: P(n,r) = n!/(n-r)!.

Interquartile range IQR = Q3 - Q1; a common outlier rule flags values below Q1 - 1.5IQR or above Q3 + 1.5IQR.

Correlation describes strength and direction of a linear association but does not prove causation.

A random sample reduces bias; a voluntary-response sample often overrepresents strong opinions.

Translate words to algebra: 'is' means equals, 'per' or 'each' suggests multiplication, 'more than' suggests addition.

Break multi-step problems into stages, solve each stage, and carry units through every computation.

Proportional reasoning: if y = kx, doubling x doubles y; find k from one known pair before predicting.

Estimate first with rounded numbers to check whether a computed answer is plausible in magnitude.

When comparing options, set up a cost or value function for each and find where they are equal (break-even point).

Interpret slope as a rate with units and intercept as a starting value in context.

Verify a solution by substituting it back into the original equation or by checking it against the real-world constraint.

Radical equations: isolate the radical, square both sides, then check for extraneous solutions.

Logarithm definition: log_b(x) = y means b^y = x; log_b(1) = 0 and log_b(b) = 1.

Logarithm product and power rules: log_b(mn) = log_b(m) + log_b(n); log_b(m^p) = p*log_b(m).

Rational expressions are simplified by factoring numerator and denominator and canceling common factors.

Direct variation y = kx versus inverse variation y = k/x; k is found from a known pair of values.

A piecewise function uses different formulas on different intervals of its domain.

Optimization steps: write the quantity as a function of one variable, differentiate, set to zero, and check endpoints.

On a closed interval, absolute extrema occur at critical points or endpoints; evaluate all candidates.

Related rates: differentiate a relationship with respect to time t, treating variables as functions of t.

Area under a curve between a and b is approximated by summing rectangles; the definite integral is the limit of that sum.

Units of a derivative are output units per input unit; units of an integral are output units times input units.

Marginal cost or revenue is the derivative of total cost or revenue with respect to quantity.

Two-way tables: joint probability is a cell count divided by the total; marginal probability uses a row or column total.

Union of two events uses the addition rule; intersection uses the multiplication rule with conditional probability.

Sample space lists all possible outcomes; the sum of probabilities over a complete sample space is 1.

The mean shifts with extreme values; the median is the 50th percentile and splits ordered data in half.

A z-score z = (x - mean)/standard deviation states how many standard deviations a value lies from the mean.

A residual is observed minus predicted value; a good linear fit has residuals scattered randomly around zero.

Sampling variability means different samples from the same population give different statistics.

Model the situation with a function, state assumptions, then test predictions against the original data.

Linear depreciation subtracts a fixed amount per period; exponential decay multiplies by a fixed factor each period.

Half-life formula: amount = initial*(1/2)^(t/half-life), where t and half-life share the same time unit.

Rounding should happen at the end of a calculation; premature rounding can distort a final answer.

Reverse-check by asking what input produces the computed output, confirming the equation was set up correctly.

Sensitivity check: alter one input slightly and see whether the conclusion changes materially.

Distance formula between two points: d = sqrt[(x2 - x1)^2 + (y2 - y1)^2].

Midpoint of a segment: ((x1 + x2)/2, (y1 + y2)/2).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is -1.

Standard form of a circle: (x - h)^2 + (y - k)^2 = r^2 with center (h,k) and radius r.

Arithmetic sequence: a_n = a_1 + (n-1)d; geometric sequence: a_n = a_1*r^(n-1).

Sum of an arithmetic series: S_n = n*(a_1 + a_n)/2.