Praxis Algebra I key terms

A symbol, usually a letter, that stands for an unknown or changing number in an expression, equation, or function. Its value may be determined by solving or may vary freely.

The numerical factor multiplied by a variable in an algebraic term. In 7x the coefficient is 7; in -3x²y it is -3. An unwritten coefficient is understood to be 1.

A quantity whose value does not change within a given expression or equation. In 4x + 9, the number 9 is the constant term because it contains no variable.

A mathematical phrase combining numbers, variables, and operations without an equals sign. Examples include 3x + 5 and (a − b)². It can be evaluated or simplified but not solved.

A statement that two expressions are equal, connected by an equals sign. Solving an equation means finding every value of the variable that makes the statement true.

A statement comparing two expressions using <, >, ≤, or ≥. Its solution is a range of values, often graphed on a number line, and the direction reverses when both sides are multiplied or divided by a negative.

A single part of an expression separated by plus or minus signs. A term consists of a coefficient and zero or more variables raised to powers, such as 5x³ or −2.

Terms that have exactly the same variable factors with the same exponents, differing only in coefficient. For example, 3x² and −8x² are like terms and can be combined into −5x².

To rewrite an expression in an equivalent but more compact form by combining like terms, applying the distributive property, and reducing fractions, without changing the expression's value.

Expressions that produce the same value for every permitted value of their variables. For instance, 2(x + 3) and 2x + 6 are equivalent because one can be transformed into the other.

The rule a(b + c) = ab + ac, allowing a factor to be multiplied across each term inside parentheses. It also underlies factoring in reverse, turning ab + ac into a(b + c).

The agreed sequence for evaluating expressions: grouping symbols, exponents, multiplication and division left to right, then addition and subtraction left to right. It ensures every expression has one standard value.

A number written above and to the right of a base showing how many times the base is used as a factor. In 2⁵ the exponent 5 means multiply five factors of 2.

In a power such as bⁿ, the quantity b that is repeatedly multiplied. In 3⁴ the base is 3 and the exponent is 4; in exponential functions the base is the constant growth factor.

A product of repeated factors, written with a base and an exponent, such as x⁵. The word also refers to the exponent itself, as in 'raised to the third power'.

A sum of terms in which every variable has a whole-number exponent and no variable appears in a denominator. Polynomials are named by degree and number of terms, such as a quadratic trinomial.

A polynomial with exactly one term, formed as the product of a constant and variables with nonnegative integer exponents. Examples include 6x², −y, and 12.

The greatest exponent found on any single variable after the polynomial is simplified, or the greatest sum of exponents within one term. It indicates the maximum number of real roots possible.

The coefficient of the term with the highest degree when a polynomial is written in descending order of exponents. Its sign predicts the end behavior of the graph.

Rewriting a polynomial as a product of simpler polynomials or factors. Common strategies include greatest common factor, difference of squares, and finding two numbers that multiply to one value and add to another.

If the product of two or more factors equals zero, then at least one factor must equal zero. This property converts a factored equation such as (x − 3)(x + 5) = 0 into simple linear equations.

An equation of the form ax² + bx + c = 0 with a ≠ 0. It can be solved by factoring, completing the square, using the quadratic formula, or reading x-intercepts from a graph.

The formula x = (−b ± √(b² − 4ac)) / (2a) giving the solutions of ax² + bx + c = 0. It works for every quadratic equation, including those that cannot be factored over the integers.

The expression b² − 4ac taken from the quadratic formula. Its sign tells how many real solutions exist: positive gives two, zero gives one, and negative gives none.

The measure of steepness of a line, calculated as the change in y divided by the change in x between two points. It represents the constant rate of change of a linear relationship.

The equation y = mx + b, where m is the slope and b is the y-intercept. It makes graphing easy because the starting point and direction are both visible in the equation.

The point where a graph crosses the y-axis, occurring where x = 0. In a real-world linear model it often represents the starting value before any change occurs.

The point where a graph crosses the x-axis, occurring where y = 0. It is also called a zero or root of the function and often represents a break-even or starting time in context.

A function whose graph is a straight line, written f(x) = mx + b, with constant rate of change. Its domain is all real numbers and it has exactly one x-intercept unless it is horizontal.

How much one quantity changes relative to another, computed as change in output divided by change in input. For a linear function it is constant and equals the slope.

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 where the graphs intersect.

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

A technique for solving systems by adding or subtracting equations so one variable cancels. Equations are often multiplied by constants first so the coefficients of one variable become opposites.

A rule assigning each input in the domain exactly one output in the range. Passing the vertical line test on a graph confirms that a relation is a function.

The complete set of input values for which a function is defined. For a real-world model it may be restricted to nonnegative numbers; for a rational function, values making a denominator zero are excluded.

The set of all possible output values a function actually produces as its input varies over the domain. On a graph it is read from the lowest to highest y-values.

A graphical check stating that a relation is a function only if no vertical line crosses its graph more than once. Two points sharing an x-value but differing in y fail the test.

The U-shaped curve that is the graph of a quadratic function. It opens upward when the leading coefficient is positive and downward when it is negative, with a vertex at its highest or lowest point.

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

The vertical line through a parabola's vertex that divides the graph into mirror-image halves. Its equation is x = −b/(2a) for a quadratic written in standard form.