CLEP College Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Slope formula

For two points (x1, y1) and (x2, y2), slope m equals (y2 − y1)/(x2 − x1). A horizontal line has slope 0; a vertical line has undefined slope.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac tells how many real roots exist.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals −1. Parallel lines instead share the same slope.

Function definition

A function assigns exactly one output to each input in its domain. A relation that gives two different outputs for the same input is not a function.

Logarithm definition

The equation y = log_b x means bʸ = x, with b > 0 and b ≠ 1. The logarithm answers the question of what exponent on b produces x.

Exponential growth model

A quantity modeled by y = a·bˣ grows when b > 1 and decays when 0 < b < 1. The value a is the starting amount at x = 0.

Combination formula

The number of unordered selections of r objects from n distinct objects is nCr = n!/(r!(n − r)!). Use it for committees and card hands where order is irrelevant.

Permutation formula

The number of ordered arrangements of r objects chosen from n distinct objects is nPr = n!/(n − r)!. Use it when order matters, such as finishing places in a race.

Addition rule for probability

For any two events, P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes outcomes counted twice because they belong to both events.

Independent events

Events A and B are independent when P(A and B) = P(A)·P(B). Knowing one occurred does not change the probability of the other.

Conditional probability formula

The probability of A given that B occurred is P(A|B) = P(A and B)/P(B), defined when P(B) > 0. It restricts attention to the outcomes inside B.

Complement rule

For any event E, the probability that E does not occur is P(not E) = 1 − P(E). This is often the fastest route when counting failures is easier than counting successes.

Mean, median, mode

The mean is the sum divided by the count; the median is the middle ordered value; the mode is the most frequent value. The median resists extreme outliers better than the mean.

Standard deviation meaning

Standard deviation measures the typical distance of data values from their mean. Adding the same constant to every value shifts the mean but leaves the standard deviation unchanged.

z-score formula

A z-score equals (value − mean)/(standard deviation). It reports how many standard deviations a value lies above or below the mean of its distribution.

Interquartile range and outliers

The IQR equals Q3 − Q1, the spread of the middle half of ordered data. Values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR are often flagged as outliers.

Correlation coefficient range

The correlation coefficient r lies between −1 and 1. Values near 1 or −1 indicate a strong linear relationship; values near 0 indicate a weak one.

Regression line slope

The least-squares regression slope equals r·(sy/sx), and the line passes through (x̄, ȳ). It predicts the average change in y per one-unit increase in x.

Simple interest formula

Simple interest is I = Prt, where P is principal, r the annual rate as a decimal, and t the time in years. The total amount is P(1 + rt).

Compound interest formula

The amount is A = P(1 + r/n)^(nt), with n compoundings per year and t years. Increasing n raises the final amount for the same nominal rate.

Continuous compounding

When interest compounds continuously, the amount equals P·e^(rt). This is the limiting case of the compound interest formula as the number of periods per year grows without bound.

Percent change formula

Percent change equals (new − old)/old × 100%. A positive result is an increase and a negative result a decrease, always measured relative to the original value.

Rule of 72

To estimate the doubling time of a growing quantity, divide 72 by the annual percentage growth rate. At 8% per year, doubling takes about 9 years.

Circle area and circumference

A circle of radius r has circumference C = 2πr and area A = πr². Area uses the radius squared, so doubling the radius quadruples the area.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The theorem applies only to right triangles and identifies the side opposite the right angle.

Polygon interior angle sum

An n-sided polygon has interior angles summing to (n − 2)·180°. In a regular polygon each interior angle measures (n − 2)·180°/n.

Similar figure scaling

Similar figures have equal angles and proportional sides. If the side scale factor is k, areas scale by k² and volumes by k³.

Set union and intersection

The union A ∪ B contains every element in A or B; the intersection A ∩ B contains only elements in both. Two sets with empty intersection are called disjoint.

Inclusion–exclusion principle

For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The subtraction corrects for elements counted twice because they belong to both sets.

Conditional and contrapositive

A conditional p → q is false only when p is true and q is false. Its contrapositive, not q → not p, is logically equivalent to it.

Divisibility tests

A number is divisible by 3 if its digit sum is divisible by 3, and by 4 if its last two digits form a multiple of 4. Divisibility by 6 requires divisibility by 2 and 3.

GCF and LCM

The greatest common factor is the product of the lowest powers of shared prime factors. The least common multiple uses the highest power of every prime appearing in either factorization.