OSAT Middle Level/Intermediate Mathematics (225) key terms

The number that, when added to a given number, produces zero. For any real number a, the additive inverse is −a, so a + (−a) = 0.

The distance of a number from zero on the number line, always nonnegative. Written |a|, it equals a when a is nonnegative and −a when a is negative.

Any number expressible as a fraction a/b where a and b are integers and b is not zero. This includes terminating and repeating decimals, integers, and fractions.

A real number that cannot be written as a ratio of two integers. Its decimal form neither terminates nor repeats, as with the square root of 2 or pi.

A whole number in the set of positive and negative counting numbers together with zero: ..., −2, −1, 0, 1, 2, ... No fractional or decimal parts belong to this set.

The set of all rational and irrational numbers, corresponding to every point on a number line. It is closed under the usual operations of addition, subtraction, multiplication, and division by nonzero values.

A way of writing numbers as a product of a factor between 1 and 10 and a power of ten, as in 3.2 × 10^5. Useful for very large or very small quantities.

Laws governing powers: multiplying like bases adds exponents, dividing subtracts them, a power of a power multiplies them, and a negative exponent gives a reciprocal. These rules also define zero exponents.

The agreed sequence for evaluating expressions: grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

The rule that multiplication distributes over addition and subtraction: a(b + c) = ab + ac. It underlies expanding expressions and combining like terms.

Solving problems by recognizing that two quantities change at a constant ratio. Setting up equivalent ratios or cross-multiplying finds unknown values in rate, scale, and percent situations.

A rate expressed per single unit of the second quantity, such as miles per one hour or cost per one item. Found by dividing the first quantity by the second.

The ratio of the amount of increase or decrease to the original amount, expressed as a percent. Computed as (new − original) ÷ original × 100.

A function whose graph is a straight line, writable as f(x) = mx + b. The constant m gives the slope and b gives the y-intercept.

The measure of steepness of a line, equal to the change in y divided by the change in x between two points. Horizontal lines have slope zero; vertical lines have undefined slope.

How much one quantity changes relative to another, typically the change in output divided by the change in input over an interval. For a linear function it equals the slope everywhere.

Writing a function as f(x) to name the rule and its input. The expression f(3) means the output of function f when the input is 3.

The domain is the set of all allowable inputs of a function; the range is the set of all resulting outputs. Restrictions arise from division by zero or even roots of negatives.

A function of the form f(x) = ax² + bx + c with a not zero. Its graph is a parabola opening upward when a is positive and downward when a is negative.

The highest or lowest point on a parabola, located at x = −b/(2a) for f(x) = ax² + bx + c. It represents the maximum or minimum value of the quadratic function.

The vertical line through the vertex that divides a parabola into mirror-image halves. For f(x) = ax² + bx + c, its equation is x = −b/(2a).

Rewriting a polynomial as a product of simpler expressions. It is used to solve quadratic equations by setting each factor equal to zero under the zero-product property.

Terms that have identical variable parts with the same exponents, so their coefficients can be combined. For example, 3x² and −5x² are like terms and combine to −2x².

Two or more equations considered together, whose solution is the set of values satisfying all of them at once. Graphically, the solution is the point or points of intersection.

A technique for solving systems by solving one equation for a variable and replacing that variable in the other equation. It reduces the system to a single equation in one unknown.

A technique for solving systems by adding or subtracting multiples of equations so one variable cancels. The result is a single equation in one variable.

A function that reverses another, so composing them returns the original input. It exists only when each output of the original function comes from exactly one input.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². It is used to find unknown side lengths and distances.

Figures with the same shape but possibly different sizes, having congruent corresponding angles and proportional corresponding side lengths. The scale factor relates all corresponding lengths.

The constant ratio comparing corresponding lengths of two similar figures. Areas scale by the square of this factor and volumes by its cube.

The distance around a circle, computed as C = 2πr or C = πd where r is the radius and d is the diameter. It is the circle's perimeter.

The amount of flat region enclosed by a circle, given by A = πr² where r is the radius. Doubling the radius quadruples the area.

The total area of all faces or curved surfaces of a three-dimensional solid, measured in square units. Formulas combine the areas of each face or use nets.

The amount of space a three-dimensional solid occupies, measured in cubic units. Common formulas include prisms (area of base times height) and spheres (four-thirds pi r cubed).

The study of geometric figures using coordinates on a plane. Distance, midpoint, and slope formulas connect algebra to shapes and their positions.

A change in a figure's position, size, or orientation, such as a translation, rotation, reflection, or dilation. Rigid transformations preserve shape and size.

The arithmetic average of a data set, found by adding all values and dividing by the number of values. It is sensitive to extreme values called outliers.

The middle value of an ordered data set, or the average of the two middle values when the count is even. It is resistant to the influence of outliers.

A data value far from the rest of the set, often defined as lying more than 1.5 interquartile ranges below the first quartile or above the third quartile.

The difference between the third quartile and the first quartile of a data set, representing the spread of the middle half of the data. It is resistant to outliers.

A measure of how spread out data values are around the mean, equal to the square root of the variance. Larger values indicate greater variability.

A number from 0 to 1 describing how likely an event is. For equally likely outcomes it equals the number of favorable outcomes divided by the total number of outcomes.

Events whose outcomes do not affect each other. The probability that both occur equals the product of their individual probabilities.

The probability that an event occurs given that another event has already occurred, written P(A|B). It equals P(A and B) divided by P(B).

A graph plotting paired data as points on a coordinate plane, used to inspect the relationship between two variables. Patterns suggest positive, negative, or no association.

A straight line drawn through a scatter plot that best models the trend in the data. It is used to make predictions and often found by least-squares regression.

A measure of the strength and direction of a linear relationship between two variables, ranging from −1 to 1. A value near zero indicates a weak linear relationship.