Math 1315 - Test #2 Review

Comprehensive vocabulary flashcards covering linear inequalities, circles, relations, functions, piecewise functions, transformations, symmetry, difference quotients, and composite functions from Math 1315 Test #2 Review.

Linear Inequality

An inequality involving a linear algebraic expression containing variables of degree one, commonly written in forms such as $ax + b < c$, $ax + b > c$, $ax + b \le c$, or $ax + b \ge c$.

Interval Notation

A mathematical notation used to represent the solution set of an inequality along a continuous span of real numbers using parentheses for excluded endpoints and brackets for included endpoints.

Set-Builder Notation

A mathematical notation describing a set by stating the properties that its members must satisfy, typically written in the format $\{x \mid \text{condition}\}$.

Rational Inequality

An inequality that contains one or more rational expressions containing a polynomial in the numerator and a non-zero polynomial in the denominator.

Distance Formula

The algebraic formula used to determine the exact distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Midpoint Formula

The formula used to locate the exact center point $M$ of a line segment connecting endpoints $(x_1, y_1)$ and $(x_2, y_2)$: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Standard Form of a Circle

The equation $(x - h)^2 + (y - k)^2 = r^2$, which defines a circle centered at $(h, k)$ with a radius of $r$.

Function

A relation between a set of inputs and a set of permissible outputs where each input value $x$ corresponds to exactly one output value $y$.

Domain

The complete set of all possible input values or $x$-coordinates for which a relation or function is defined.

Range

The complete set of all possible resulting output values or $y$-coordinates produced by a relation or function.

Piecewise Function

A function whose rule is given by different formulas or expressions over specified sub-intervals of its domain.

Parent Functions

The simplest core functions in a family of functions, including the Identity ($y = x$), Squaring ($y = x^2$), Absolute Value ($y = |x|$), Cubing ($y = x^3$), Square Root ($y = \sqrt{x}$), Cubed Root ($y = \sqrt[3]{x}$), and Reciprocal ($y = \frac{1}{x}$) functions.

Vertical Shift

A rigid transformation that translates a graph upward or downward by adding or subtracting a constant $c$ outside the parent function: $y = f(x) + c$.

Horizontal Shift

A rigid transformation that translates a graph horizontally to the left or right by modifying the input argument: $y = f(x - c)$.

Reflection across the x-axis

A transformation of a function given by $y = -f(x)$ that flips the graph vertically across the $x$-axis.

Increasing Function

A function behavior over an interval where the graph rises from left to right, satisfying $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.

Decreasing Function

A function behavior over an interval where the graph falls from left to right, satisfying $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

Constant Function

A function behavior over an interval where output values remain identical across all inputs, satisfying $f(x_1) = f(x_2)$ for all $x_1$ and $x_2$.

Even Function

A function that satisfies the algebraic condition $f(-x) = f(x)$ for all $x$ in its domain, displaying symmetry with respect to the $y$-axis.

Odd Function

A function that satisfies the algebraic condition $f(-x) = -f(x)$ for all $x$ in its domain, displaying symmetry with respect to the origin.

Difference Quotient

The algebraic expression $\frac{f(x+h) - f(x)}{h}$ (where $h \neq 0$), which computes the average rate of change of $f(x)$ over the interval $[x, x+h]$.

Composition of Functions

The mathematical operation combining two functions where the output of one function becomes the input of another, denoted $(f \circ g)(x) = f(g(x))$.