GA Milestones Algebra EOC flashcards

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

Slope formula

For points (x1, y1) and (x2, y2), slope m equals (y2 − y1) divided by (x2 − x1). It measures vertical change per unit of horizontal change, and is undefined for vertical lines.

Slope-intercept form

The equation y = mx + b, where m is the slope and b is the y-intercept. It lets you graph a line by plotting (0, b) and moving according to slope, or read slope and intercept directly.

Point-slope form

The equation y − y1 = m(x − x1) describes a line with slope m through point (x1, y1). It is useful when two points or a point and a slope are given.

Standard form of a line

The form Ax + By = C, usually with integer coefficients and A ≥ 0. Its slope is −A/B, and intercepts are found by setting each variable to zero in turn.

Parallel and perpendicular slopes

Parallel lines have identical slopes. Perpendicular lines have slopes that are negative reciprocals, so their product is −1. Vertical lines are perpendicular to horizontal lines.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, solutions are x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic, including ones that do not factor over the integers.

Discriminant

The expression b² − 4ac from a quadratic in standard form. Positive means two distinct real solutions, zero means exactly one real solution, and negative means no real solutions.

Vertex of a parabola

For y = ax² + bx + c, the vertex x-coordinate is −b/(2a). The vertex is the minimum when a > 0 and the maximum when a < 0.

Axis of symmetry

The vertical line x = −b/(2a) that divides a parabola into mirror-image halves. Points on the parabola equidistant from this line share the same y-value.

Zero product property

If the product of factors equals zero, at least one factor must equal zero. Setting each factor of a factored polynomial to zero gives its solutions and x-intercepts.

Difference of squares

The pattern a² − b² factors as (a − b)(a + b). Recognizing it quickly factors expressions such as x² − 49 = (x − 7)(x + 7).

Greatest common factor

The largest expression dividing every term of a polynomial. Factoring it out first simplifies later steps, as in 6x³ + 9x² = 3x²(2x + 3).

Exponent product rule

When multiplying powers with the same base, add the exponents: a^m · a^n = a^(m+n). This follows from counting repeated factors and applies to numeric and variable bases.

Negative and zero exponents

For a ≠ 0, a^0 = 1 and a^(−n) = 1/a^n. Negative exponents indicate reciprocals, not negative values, so 2^(−3) equals 1/8.

Rational exponents

A fractional exponent a^(m/n) equals the nth root of a raised to m, written ⁿ√(a^m). It connects radical expressions with exponent rules and allows either root-first or power-first evaluation.

Function definition

A relation in which every input has exactly one output. Graphs of functions pass the vertical line test; a single vertical line never crosses the graph more than once.

Domain and range

Domain is the set of all valid inputs of a function; range is the set of resulting outputs. Context can restrict domain, such as excluding negative time or fractional items.

Function notation

Writing f(x) to represent the output of function f for input x. Evaluating means substituting the input, so f(3) replaces every x in the rule with 3.

Linear function rate of change

For a linear function, the rate of change is constant and equals the slope. Each one-unit increase in input produces the same change in output, producing a straight graph.

Exponential growth model

The equation y = a(1 + r)^t models growth, where a is the initial amount, r the growth rate as a decimal, and t the number of periods. Outputs change by a constant ratio.

Exponential decay model

The equation y = a(1 − r)^t models decay, where a is the initial amount and r the decay rate as a decimal between 0 and 1. Each period multiplies the previous amount by (1 − r).

System of linear equations

Two or more equations considered together; a solution satisfies all of them at once. Graphically, the solution is the point where the lines intersect.

Substitution method

Solve one equation for a variable, then substitute that expression into the other equation. The result is a single equation in one variable, which is then solved.

Elimination method

Add or subtract multiples of the equations so one variable cancels. Solve the remaining equation, then substitute back to find the other variable.

Number of solutions in systems

Intersecting lines give one solution; parallel lines with different intercepts give none; identical lines give infinitely many. Comparing slopes and intercepts identifies the case.

Absolute value equation

An equation of the form |ax + b| = c with c ≥ 0 splits into ax + b = c and ax + b = −c. If c is negative, the equation has no solution.

Extraneous solution

A value produced during solving that does not satisfy the original equation. It arises from squaring both sides or clearing variable denominators, so answers must be checked.

Arithmetic sequence formula

The nth term is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d the common difference. Consecutive terms differ by the same constant.

Geometric sequence formula

The nth term is aₙ = a₁ · r^(n−1), where a₁ is the first term and r the common ratio. Each term equals the previous term multiplied by r.

Line of best fit

A line modeling the trend of a scatter plot, used to predict values. It may be estimated by eye or found by regression, and predictions outside the data range are extrapolations.

Correlation coefficient

A value from −1 to 1 describing the strength and direction of a linear relationship. Values near 1 or −1 are strong; values near 0 are weak.

Correlation versus causation

A strong correlation between two variables does not prove one causes the other. A lurking or confounding variable may drive both, so conclusions require study design evidence.