Algebra and Geometry Fundamentals Flashcards

Vocabulary flashcards covering algebra rules, equations, inequalities, factorials, functions, domain and range, and geometric formulas.

Imaginary Unit $i$ Powers

The imaginary unit is defined as $i = \sqrt{-1}$, with higher powers given by $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$.

Exponent Product Rule

The rule stating that $x^m \times x^n = x^{m+n}$.

Exponent Quotient Rule

The rule stating that \frac{x^m}{x^n} = x^{m-n}.

Exponent Power Rule

The rule stating that $(x^m)^n = x^{mn}$.

Approximation of $\sqrt{2}$

$$\sqrt{2} \approx 1.4$$

Approximation of $\sqrt{3}$

$$\sqrt{3} \approx 1.7$$

Approximation of $\sqrt{5}$

$$\sqrt{5} \approx 2.2$$

Slope-Intercept Form

The equation of a line given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Point-Slope Form

The equation of a line given by $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a given point.

Slope Formula

The slope $m$ between two points is $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$.

Parallel Lines Slope Condition

Two lines are parallel if their slopes are equal, $m_1 = m_2$.

Perpendicular Lines Slope Condition

Two lines are perpendicular if their slopes are negative reciprocals, $m_1 = -\frac{1}{m_2}$.

Horizontal Line Equation and Slope

A horizontal line has equation $y = 4$ (or $y = c$) and slope $m = 0$.

Vertical Line Equation and Slope

A vertical line has equation $x = 4$ (or $x = c$) and an undefined slope.

X-Intercept Procedure

Let $y = 0$ and solve the equation for $x$.

Y-Intercept Procedure

Let $x = 0$ and solve the equation for $y$.

Quadratic Formula

The roots of $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Distance Formula

The distance $D$ between two points is $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Absolute Value Radical Identity

$$\sqrt{a^2} = |a|$$

Algebraic Definition of Absolute Value

$|x|$ equals $-x$ if $x < 0$, $0$ if $x = 0$, and $x$ if $x > 0$.

Factorials ($0!$ through $5!$)

$0! = 1$, $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, and $5! = 120$.

Equal Functions Condition

Functions $f(x)$ and $g(x)$ are equal if and only if their domains are identical ($D_f = D_g$) and $f(x) = g(x)$ for all $x$.

Composite Functions

$(f \circ g)(x) = f(g(x))$ and $(g \circ f)(x) = g(f(x))$.

Three Rules for Finding Domain

1) No $0$ in denominator, 2) No negative numbers under a square root ($\sqrt{\text{negative}}$), 3) No numbers less than or equal to $0$ inside a logarithm ($\log(\le 0)$).

Two Ways to Find Range

1) Graph the function $y = f(x)$, 2) Find the domain of the inverse function.

Two Steps to Find Inverse

1) Swap $x$ and $y$, 2) Solve the resulting equation for $y$.

Even Function Definition and Symmetry

An even function satisfies $f(-x) = f(x)$ and possesses y-axis symmetry.

Odd Function Definition and Symmetry

An odd function satisfies $f(-x) = -f(x)$ and possesses origin symmetry.

Circumference of a Circle

$C_{\text{circle}} = 2\pi r$ or $C_{\text{circle}} = \pi d$.

Area of a Circle

$A_{\text{circle}} = \pi r^2$.

Surface Area of a Sphere

$SA_{\text{sphere}} = 4\pi r^2$.

Area of a Triangle

$A_{\text{triangle}} = \frac{1}{2}bh$.

Area of a Trapezoid

$A_{\text{trapezoid}} = \frac{1}{2}h(b_1 + b_2)$.

Volume of a Cylinder

$V_{\text{cylinder}} = \pi r^2 h$.

Volume of a Cone

$V_{\text{cone}} = \frac{1}{3}\pi r^2 h$.

Volume of a Sphere

$V_{\text{sphere}} = \frac{4}{3}\pi r^3$.

Volume of a Rectangular Solid

$V_{\text{rect. solid}} = lwh$.