Math

Function

A relation where each individual input (x) maps to exactly one unique output (y).

Domain

The complete set of all possible independent input values (x-values) for a relation or function.

Range

The complete set of all possible dependent output values (y-values) for a relation or function.

Vertical Line Test

A visual method used to determine if a graph represents a function; the graph is a function if no vertical line crosses it more than once.

f(x) notation

A mathematical shorthand read as "f of x" used to represent the output (y-value) of a function f for a given input x.

If f(x) = 3x - 5, what is f(4)?

7 (Calculated as 3(4) - 5 = 12 - 5)

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁), representing the ratio of the vertical change to the horizontal change (rise over run).

Rate of Change

A ratio that compares how much a dependent variable changes relative to a change in the independent variable.

Slope of a horizontal line

Zero (0), because there is no vertical change (rise = 0).

Slope of a vertical line

Undefined, because there is no horizontal change (run = 0, dividing by zero is impossible).

Linear Function

A function whose graph forms a straight line and has a constant rate of change.

Slope-Intercept Form

y = mx + b, where m represents the slope and b represents the y-intercept.

Point-Slope Form

y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a specific point on the line.

Standard Form of a linear equation

Ax + By = C, where A, B, and C are integers, and A must be greater than or equal to zero.

y-intercept

The point where a line crosses the y-axis, occurring when the input x equals 0.

x-intercept

The point where a line crosses the x-axis, occurring when the output y (or f(x)) equals 0.

Parallel Lines

Lines in the same plane that never intersect and have the exact same slope.

Perpendicular Lines

Lines that intersect to form right angles and have slopes that are negative reciprocals of each other.

Absolute Value Function Parent

f(x) = |x|, which forms a distinct V-shape symmetric across the y-axis.

Vertex of an Absolute Value Function

The highest or lowest turning point on a V-shaped graph; for f(x) = |x|, the vertex is at (0,0).

f(x) = |x| + k

A vertical translation that shifts the parent absolute value graph UP by k units.

f(x) = |x| - k

A vertical translation that shifts the parent absolute value graph DOWN by k units.

f(x) = |x - h|

A horizontal translation that shifts the parent absolute value graph RIGHT by h units.

f(x) = |x + h|

A horizontal translation that shifts the parent absolute value graph LEFT by h units.

f(x) = -|x|

A reflection over the x-axis that flips the V-shape upside down.