Algebra–Precalculus Review Guide (Limits Excluded)

Vocabulary flashcards covering key concepts from Algebra–Precalculus Review Guide (Limits Excluded)

Function notation f(x)

Notation for a function that maps inputs to outputs; evaluate at numbers or expressions.

Slope-intercept form

y = mx + b; a line form where m is the slope and b is the y-intercept.

Point-slope form

y - y1 = m(x - x1); line form using slope m and a point (x1, y1).

Standard form (line)

Ax + By = C; linear equation form with coefficients A, B, C.

Parallel lines

Lines with the same slope; they never intersect.

Perpendicular lines

Lines whose slopes are negative reciprocals; they intersect at a right angle.

Quadratic standard form

y = ax^2 + bx + c; standard form for a parabola.

Quadratic vertex form

y = a(x - h)^2 + k; vertex is (h, k).

Vertex

The minimum or maximum point of a parabola; for y = a(x - h)^2 + k it is (h, k).

Quadratic formula

x = (-b ± sqrt(b^2 - 4ac)) / (2a); solves ax^2 + bx + c = 0.

Opens upward/downward

Parabola opens upward if a > 0; downward if a < 0.

Domain

Set of x-values for which a function is defined.

Range

Set of possible output values y of a function.

Vertical asymptote

x-values where a function grows without bound due to denominator zero (and not canceled).

Horizontal asymptote

y-value approached as x → ±∞ for rational functions; if deg(numerator) < deg(denominator) then y = 0; if equal degrees then leading coefficients ratio; if greater, none (may have a slant).

Slant (oblique) asymptote

An oblique asymptote that occurs when degree(numerator) = degree(denominator) + 1 in rational functions.

Radical ↔ exponent

n-th root: x^(m/n) equals sqrt[n]{x^m}.

Negative exponent

a^-n = 1/(a^n); reciprocal.

Log product rule

logb(MN) = logb M + log_b N.

Log quotient rule

logb(M/N) = logb M - log_b N.

Log power rule

logb(M^p) = p logb M.

Change of base formula

log_b(M) = ln(M)/ln(b); convert logs to natural or common logs.

Unit circle

A circle of radius 1 used to find trig values; key coordinates relate to cos and sin.

π/6, π/4, π/3 values

At angles π/6, π/4, π/3: cos,sin values are (√3/2, 1/2), (√2/2, √2/2), (1/2, √3/2) respectively.

Quadrantal angles

0, π/2, π, 3π/2 correspond to (1,0), (0,1), (-1,0), (0,-1).

Signs by quadrant

I: all positive; II: sin positive, cos negative, tan negative; III: sin negative, cos negative, tan positive; IV: sin negative, cos positive, tan negative.

Exponential equation solving

Isolate the exponent and take natural logs: a^x = b → x = ln(b)/ln(a).

Logarithmic equation solving

Rewrite as exponential form: log_b(x) = c → x = b^c (or for ln, x = e^c).

Factorization (polynomials)

Factor polynomials to solve equations; e.g., x^2 - 3x = 0 factors as x(x - 3) = 0.

Higher powers factoring

Factoring higher-degree polynomials (by grouping or patterns) to solve.

Inverse function

A function that undoes another; to find the inverse, solve for x after swapping x and y (f^{-1}).

Composition of functions

(f∘g)(x) = f(g(x)); apply g first, then f.

Sum of functions

(f+g)(x) = f(x) + g(x).