Mathematics: Functions, Algebra, Analytical Geometry, and Progressions

Comprehensive vocabulary flashcards covering functions parity, domain restrictions, progressions, analytic geometry formulas, quadratic equations, and logarithmic rules.

Even Function

A function whose domain is symmetric with respect to zero and satisfies the condition $f(-x) = f(x)$. Its graph is symmetric across the vertical $y$-axis (axis of ordinates). Examples include $y = x^2$, $y = \cos(x)$, and $y = |x|$.

Odd Function

A function whose domain is symmetric with respect to zero and satisfies the condition $f(-x) = -f(x)$. Its graph is symmetric with respect to the origin $(0, 0)$. Examples include $y = x^3$, $y = \sin(x)$, and $y = \frac{1}{x}$.

Function of General Form

A function that is neither even nor odd because the required symmetry equality ($f(-x) = f(x)$ or $f(-x) = -f(x)$) does not hold.

Slope Formula

The ratio determining the steepness and direction of a line through points $(x_1, y_1)$ and $(x_2, y_2)$, defined as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Rational Function Domain Restriction

For a rational function of the form $y = \frac{f(x)}{g(x)}$, the denominator cannot equal zero, establishing the domain restriction $g(x) \neq 0$.

Radical Function Domain Restriction

For an even root function of the form $y = \sqrt{g(x)}$, the expression under the radical must be non-negative, requiring $g(x) \ge 0$.

Logarithmic Function Domain Restriction

For a logarithmic expression of the form $y = \log_b(g(x))$, the argument must be strictly positive, requiring $g(x) > 0$.

Vieta's Formulas (Monic Quadratic)

Relations between the roots and coefficients of a quadratic equation $x^2 + px + q = 0$, where the sum of the roots is $x_1 + x_2 = -p$ and the product of the roots is $x_1 \cdot x_2 = q$.

Arithmetic Progression ($n$-th Term)

The formula to calculate the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$, where $d = a_2 - a_1$ is the common difference.

Arithmetic Progression (Sum of First $n$ Terms)

The formula representing the sum of the first $n$ terms of an arithmetic progression: $S_n = \frac{a_1 + a_n}{2} \cdot n$.

Geometric Progression ($n$-th Term)

The formula to calculate the $n$-th term of a geometric sequence: $b_n = b_1 \cdot q^{n - 1}$, where $q = \frac{b_2}{b_1}$ is the common ratio.

Geometric Progression (Finite Sum)

The formula for the sum of the first $n$ terms of a geometric progression: $S_n = \frac{b_1(q^n - 1)}{q - 1} = \frac{b_1(1 - q^n)}{1 - q}$.

Infinite Decreasing Geometric Progression Sum

The limiting sum of an infinitely decreasing geometric sequence when the common ratio satisfies $|q| < 1$, evaluated as $S = \frac{b_1}{1 - q}$.

Distance Formula Between Two Points

The Cartesian coordinate formula for calculating the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Equation of a Circle

The standard form equation of a circle with center $(x_0, y_0)$ and radius $r$, given by $(x - x_0)^2 + (y - y_0)^2 = r^2$, which reduces to $x^2 + y^2 = r^2$ when centered at the origin.

Horizontal Parabola (Standard Equation)

The standard algebraic equation of a parabola opening along the horizontal axis: $y^2 = 4px$.

Logarithm Product Rule

The property stating that the logarithm of a product equals the sum of the logarithms: $\log_a(mn) = \log_a(m) + \log_a(n)$.

Logarithm Quotient Rule

The property stating that the logarithm of a quotient equals the difference of the logarithms: $\log_a\left(\frac{m}{n}\right) = \log_a(m) - \log_a(n)$.

Logarithm Power Rule

The property allowing an exponent within a logarithmic argument to be brought out as a coefficient: $\log_a(m^k) = k \cdot \log_a(m)$.

Logarithmic Change of Base Formula

The identity allowing a logarithm to be converted to a new base: $\log_c(b) = \frac{\log_a(b)}{\log_a(c)}$, or in terms of natural logarithms, $\log_a(b) = \frac{\ln(b)}{\ln(a)}$ for $a \neq 1$ and $b > 0$.

Complex Numbers ($\mathbb{C}$)

The set of numbers of the form $a + bi$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit.