NYSTCE Math CST flashcards

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

Polya's four problem-solving phases

Understand the problem, devise a plan, carry out the plan, and look back to verify. Each phase produces a checkable result, and verification is a required step rather than optional.

Counterexample

A single case in which a general statement is false. One counterexample disproves a universal claim, while confirming examples alone can never establish that a claim is true.

Dimensional analysis

A method treating units as algebraic quantities that cancel across multiplication and division. It converts rates, checks whether an equation is consistent, and catches order-of-magnitude errors.

Rational vs irrational numbers

Rational numbers can be written as a ratio of integers and have terminating or repeating decimals. Irrational numbers, such as the square root of 2 or pi, have nonrepeating, nonterminating decimals.

Greatest common divisor

The largest integer dividing two or more numbers evenly. From prime factorizations, multiply the lowest power of each prime common to all the numbers.

Least common multiple

The smallest positive integer divisible by each of the given numbers. From prime factorizations, multiply the highest power of every prime that appears in any number.

Negative and fractional exponents

A negative exponent gives the reciprocal of the base raised to the positive exponent. A fractional exponent one over n denotes the nth root of the base.

Conjugate of a complex number

For a + bi the conjugate is a − bi. Multiplying numerator and denominator by the conjugate removes i from the denominator because the product (a + bi)(a − bi) equals a squared plus b squared.

Successive percent changes

Percent changes compound multiplicatively rather than adding. A 20% increase followed by a 20% decrease multiplies by 1.2 and 0.8, giving 0.96, a net 4% decrease.

Absolute value as distance

The expression |x − a| gives the distance between x and a on the number line. The inequality |x − a| < d is equivalent to a − d < x < a + d.

Perpendicular and parallel slopes

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is negative one, so each slope is the negative reciprocal of the other.

Quadratic discriminant

The quantity b squared minus 4ac in the quadratic formula. Positive gives two distinct real roots, zero gives one repeated real root, negative gives two complex conjugate roots.

Vertex of a parabola

For y = ax squared + bx + c the vertex lies at x = negative b divided by 2a. The parabola opens upward when a is positive and downward when a is negative.

Compound interest formula

A = P(1 + r/n)^(nt), where P is principal, r the annual rate as a decimal, n the compounding periods per year, and t the years. Continuous compounding uses Pe^(rt).

Inverse function condition

A function has an inverse only when it is one-to-one, meaning each output comes from exactly one input. The graph of an inverse reflects the original graph across the line y = x.

Change of base formula

log base b of x equals ln x divided by ln b. This converts any logarithm to natural or common logarithms so it can be evaluated with a calculator.

Function transformations

Replacing x with x − h shifts the graph right h units, adding k shifts it up k, negating the function reflects over the x-axis, and multiplying by a scales vertically.

Interior angle sum of a polygon

A polygon with n sides has interior angles summing to (n − 2) times 180 degrees. For a regular polygon, divide that sum by n to get each interior angle.

Similar figure scaling

When two figures are similar with side ratio k, corresponding angles are equal, areas scale by k squared, and volumes scale by k cubed.

Sector area formula

A sector with central angle theta degrees in a circle of radius r has area (theta/360) times pi r squared. Its arc length is (theta/360) times 2 pi r.

Volume of a sphere

The volume of a sphere of radius r is four-thirds pi r cubed, and its surface area is 4 pi r squared. Both formulas require the radius, not the diameter.

Distance formula

The distance between two points equals the square root of the sum of the squared differences of their x-coordinates and y-coordinates. It follows directly from the Pythagorean theorem.

Special right triangles

In a 45-45-90 triangle the sides are x, x, and x times the square root of 2. In a 30-60-90 triangle they are x, x times the square root of 3, and 2x.

Law of cosines

c squared equals a squared plus b squared minus 2ab times the cosine of angle C. It solves triangles when two sides and the included angle, or all three sides, are known.

Mean versus median

The mean is the sum divided by the count and is pulled toward outliers and skew. The median is the middle of the ordered data and resists extreme values.

Standard deviation scaling

Adding a constant to every data value leaves the standard deviation unchanged because spread is preserved. Multiplying every value by k multiplies the standard deviation by the absolute value of k.

Addition rule for probability

For two events, the probability of A or B equals the probability of A plus the probability of B minus the probability of both occurring, correcting for outcomes counted twice.

Conditional probability

The probability of A given B equals the probability of both A and B divided by the probability of B. It restricts the sample space to outcomes where B has occurred.

Permutations versus combinations

Permutations count ordered arrangements using n factorial divided by (n minus r) factorial. Combinations count unordered selections and divide that result further by r factorial.

Empirical rule

For a normal distribution roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.

Correlation coefficient

A number from negative one to positive one describing the strength and direction of linear association. It measures linear relationship only and does not establish causation.

Residual

The difference between an observed value and the value predicted by a regression line. A residual plot showing a curved pattern indicates that a linear model is inappropriate.