GA Milestones Algebra EOC cheat sheet

Slope between two points: m = (y2 − y1)/(x2 − x1). Slope-intercept form y = mx + b gives slope m and y-intercept b; point-slope form y − y1 = m(x − x1) uses any known point.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so m1 · m2 = −1. A vertical line has undefined slope and equation x = a; a horizontal line has slope 0 and equation y = b.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) for ax² + bx + c = 0. The discriminant b² − 4ac gives two real roots if positive, one real root if zero, and no real roots if negative.

Vertex of a parabola y = ax² + bx + c occurs at x = −b/(2a); the vertex is a maximum when a < 0 and a minimum when a > 0. Factored form y = a(x − r)(x − s) shows x-intercepts r and s.

Exponent rules: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n; a^(1/n) = ⁿ√a.

Systems of two linear equations: substitution isolates one variable; elimination adds or subtracts multiples of equations. A system has no solution when slopes are equal but intercepts differ, and infinitely many solutions when the equations describe the same line.

Function notation: f(x) names the output for input x. A relation is a function only if each input has exactly one output; the vertical line test confirms this on a graph. Domain is the set of inputs, range the set of outputs.

Exponential growth and decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where a is the initial value and r the rate per period. Linear models change by a constant difference; exponential models change by a constant ratio.

Slope formula m = (y2 − y1)/(x2 − x1) underlies rate-of-change questions; in context, slope is the change in output per one-unit change in input, and the y-intercept is the starting value.

Standard form Ax + By = C can be rewritten as y = −(A/B)x + C/B, so its slope is −A/B. Horizontal lines y = b and vertical lines x = a are the two special cases that fail the slope formula.

Factoring strategies in order: greatest common factor first, then difference of squares a² − b² = (a − b)(a + b), then trinomials such as x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

Zero product property: if a product equals zero, at least one factor is zero. Solve factored equations by setting each factor to zero; this yields the x-intercepts of the related graph.

Radical and rational equations require checking for extraneous solutions: squaring both sides or multiplying by a variable expression can introduce values that fail the original equation.

Arithmetic sequences add a constant difference d, with aₙ = a₁ + (n − 1)d; geometric sequences multiply by a constant ratio r, with aₙ = a₁ · r^(n−1).

Absolute value equations |ax + b| = c with c ≥ 0 split into ax + b = c and ax + b = −c; if c < 0 there is no solution. Graphs of absolute value functions form a V with vertex where the expression equals zero.

Data displays: a line of best fit models scatter-plot trend, and correlation near 1 or −1 indicates a strong linear relationship while near 0 indicates weak linear relationship. Correlation does not establish causation.