Study Guide for Linear and Quadratic Equations, Functions, and Inequalities

Vocabulary flashcards covering linear and rational equations, complex numbers, quadratic formulas and discriminant, inequalities, linear forms, functions, business functions, and parabolas.

Linear Equation

An equation in which the highest exponent of the variable is $1$.

Rational Equations

Equations that involve fractions with polynomial expressions in the numerator and denominator.

Restrictions on the Variable

Values that make any denominator zero in a rational equation.

Imaginary Unit

Denoted as $i$, defined such that $i^2 = -1$.

Complex Conjugates

Expressions of the form $(a + bi)(a - bi)$, which simplify to the product $a^2 + b^2$.

Quadratic Formula

The formula x = \frac{-b \ne \b \t \r \n  \f \pm \sqrt{b^2 - 4ac}}{2a} used to solve quadratic equations.

Discriminant

Given by $D = b^2 - 4ac$, which determines the nature of the roots of a quadratic equation: if $D > 0$, there are two distinct real roots; if $D = 0$, there is one repeated real root.

Linear Inequalities

Mathematical statements similar to linear equations that utilize inequality symbols ($>$, $<$, $\ge$, $\le$), whose solutions can be represented on a number line.

Compound Inequalities

Mathematical statements involving two inequalities connected by "and" or "or", where each part is solved separately.

Absolute Value Inequalities

Inequalities solved by considering two cases for the expression inside the absolute value: (a) without the absolute value, and (b) negating the expression.

Point-Slope Form

An equation of a line passing through a given point $(x_1, y_1)$ with slope $m$, expressed as $y - y_1 = m(x - x_1)$.

Slope Formula (Two Points Form)

The formula used to determine the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$, given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope-Intercept Form

An equation of a line written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Parallel Lines

Lines that share the exact same slope.

Perpendicular Lines

Lines whose slopes are negative reciprocals of one another, satisfying $m_2 = -\frac{1}{m_1}$.

Piecewise Functions

Functions in which the specific piece of the function to be used is determined by the input value.

Domain of Functions

The allowable set of input values found by identifying restrictions upon values of $x$ from fractions and square roots.

Cost Function ($C$)

A function that represents the total cost of production.

Revenue Function ($R$)

A function that represents the total income generated from sales.

Profit Function ($P$)

The difference between total revenue and total cost, calculated using the equation $P = R - C$.

Break-Even Points

The intersection points determined by setting the cost function equal to the revenue function ($C = R$).

Vertex Form of a Parabola

The quadratic representation $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.

Orientation of a Parabola

The direction in which a parabola opens based on coefficient $a$: it opens upward when $a > 0$ (having a minimum value at the vertex) and opens downward when $a < 0$ (having a maximum value at the vertex).