LCA Mathematical Applications key terms

A number that can be written as a fraction of two integers, with a nonzero denominator. Its decimal form either ends or repeats a fixed block of digits forever.

A real number that cannot be written as a fraction of two integers. Its decimal form never ends and never settles into a repeating block, as with the square root of two.

Any value on the continuous number line, covering both rational and irrational numbers. Real numbers can be compared, ordered, and combined with the usual arithmetic operations.

A whole number with no fractional part, including positive values, negative values, and zero. Integers are closed under addition, subtraction, and multiplication but not always under division.

The distance of a number from zero on the number line, always zero or positive. It is written with vertical bars around the value and removes any negative sign.

A way of writing very large or very small numbers as a coefficient between one and ten multiplied by a power of ten. It makes the scale of a quantity easy to compare.

A rough measure of size expressed as the nearest power of ten. Two quantities differ by one order of magnitude when one is about ten times the other.

A small raised number showing how many times a base is used as a factor. Rules for combining exponents cover multiplication, division, powers of powers, and negative or zero values.

A value that produces a given number when multiplied by itself. Every positive number has two square roots, one positive and one negative, and the radical symbol usually means the positive one.

A comparison of two quantities by division, often written as a fraction or with a colon. Ratios describe relative size rather than absolute counts.

A statement that two ratios are equal. Solving a proportion means finding the unknown value that keeps the two ratios in the same relative balance.

A rate expressed per single unit of the second quantity, such as cost per item or distance per hour. It allows direct comparison between different sized offers or situations.

A function whose graph is a straight line and whose rule can be written as a constant times the input plus a constant. It changes by equal amounts for equal input steps.

The measure of steepness of a line, found as the change in output divided by the change in input between two points. It gives the constant rate of change of a linear function.

The output value where a graph crosses the vertical axis, occurring when the input equals zero. In applied problems it often represents a starting amount or fixed cost.

How much one quantity changes relative to a change in another. For a linear relationship it is constant and equals the slope; for other functions it varies across the graph.

A function whose rule includes the square of the input as its highest power, producing a U-shaped graph called a parabola. It has a single maximum or minimum point called the vertex.

The turning point of a parabola, where the graph reaches its highest or lowest output value. It marks the axis of symmetry and often answers optimization questions in context.

A function in which the input appears in the exponent, so the output is multiplied by a fixed factor for each equal input step. It models growth or decay at a constant percent rate.

The multiplier applied to a quantity in each time period of exponential change. A factor above one indicates growth, while a factor between zero and one indicates decay.

A set of two or more equations that must hold true at the same time. A solution is the set of values satisfying every equation, found graphically or by substitution and elimination.

A statement comparing two expressions with symbols such as less than, greater than, or at least. Its solution is a range of values rather than a single number.

A compact way of naming a function and its output, such as f of x, where the symbol in parentheses is the input. It makes substitution and evaluation explicit.

The domain is the full set of allowed input values for a function; the range is the full set of resulting output values. In context, both are often restricted to realistic quantities.

The greatest value a quantity or function attains over a stated interval or situation. Finding it is central to optimization problems such as maximizing profit or area.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. It links side lengths and is used to find unknown distances.

A triangle containing one angle of exactly ninety degrees. Its longest side, opposite the right angle, is the hypotenuse, and the other two sides are the legs.

A ratio of two side lengths in a right triangle, named sine, cosine, or tangent depending on which sides are compared. These ratios connect an acute angle to side lengths.

Shapes with the same form but possibly different size, having equal corresponding angles and proportional corresponding side lengths. Their areas relate by the square of the scale factor.

The constant multiplier relating corresponding lengths of two similar figures or a scale drawing to the real object. Area and volume scale by its square and cube respectively.

The amount of three-dimensional space an object occupies, measured in cubic units. Formulas depend on the shape, such as length times width times height for a rectangular solid.

The total area of all outer faces of a three-dimensional object, measured in square units. It answers questions about covering, painting, or wrapping a shape.

A rule giving the straight-line distance between two points on a coordinate plane, derived from the Pythagorean theorem. It uses the horizontal and vertical separations of the points.

The arithmetic average of a data set, found by adding all values and dividing by the count. It is sensitive to extreme values because every observation contributes.

The middle value of an ordered data set, or the average of the two middle values when the count is even. It resists distortion by unusually large or small observations.

A measure of how far data values typically sit from their mean. A small value means tightly clustered data; a large value means widely spread data.

One of three values that split ordered data into four groups of roughly equal size. The second quartile is the median, and the range between first and third describes the middle half.

A symmetric bell-shaped pattern of data in which most values cluster near the mean and grow rarer farther away. Fixed percentages of values fall within one, two, or three standard deviations.

A number from zero to one expressing how likely an event is, often as favorable outcomes divided by total equally likely outcomes. Zero means impossible and one means certain.

Two events whose outcomes do not influence each other, so the chance of both equals the product of their separate chances. Sampling with replacement usually produces independent events.

The chance that one event occurs given that another event is already known to have occurred. It restricts attention to a smaller group of outcomes.

A line drawn through a scatterplot that best summarizes the overall trend of paired data. It supports predictions and shows whether the association is positive or negative.

A measure of the strength and direction of a linear association between two variables, ranging from negative one to positive one. It does not by itself prove that one variable causes the other.

A symbol, usually a letter, standing for a quantity that can change or is not yet known. Identifying what a variable represents is the first step in translating a situation into mathematics.

A condition that limits the possible values of the variables in a model, such as a budget, a time limit, or a physical capacity. Constraints are often written as equations or inequalities.

A mathematical representation of a real situation, built from variables, relationships, and simplifying assumptions. A model is judged by how well its predictions serve the question asked.

Using a model to predict values outside the range of the observed data. Such predictions are riskier than interpolation because the pattern may not continue beyond the data.