Glossary
Domain
The set of all possible input values (x-values) for which a function or relation is defined, often restricted by square roots of negative numbers or division by zero.
Example:
For the function y = √(x - 2), the domain is x ≥ 2 because the expression under the square root cannot be negative.
Explicit Functions
Functions where the dependent variable (y) is directly expressed or isolated in terms of the independent variable (x), typically in the form y = f(x).
Example:
y = 3x² - 5 is an explicit function where y is clearly defined by x.
Graph (of an equation with x and y)
The visual representation of all ordered pairs (x, y) that satisfy a given equation, showing the geometric relationship between the variables.
Example:
The graph of x² + y² = 9 is a circle centered at the origin with a radius of 3.
Horizontal Interval
A segment of a graph where the y-value remains constant while the x-value changes, meaning the rate of change of x with respect to y (Δx/Δy) is zero, and the slope is zero.
Example:
On a roller coaster ride, a flat section where the height doesn't change as the car moves forward represents a horizontal interval.
Implicit Functions
Equations where the dependent variable (y) is not isolated and is intertwined with the independent variable (x), often requiring implicit differentiation to find the derivative.
Example:
The equation sin(xy) = x + y is an implicit function because y cannot be easily solved for in terms of x.
Implicitly Defined Functions
Equations where variables (x and y) are mixed and y is not easily isolated, describing a relationship between them rather than explicitly defining y in terms of x.
Example:
The equation x³ + y³ = 6xy defines a folium of Descartes, where y is implicitly related to x.
Negative Slope
Indicates that as the independent variable (x) increases, the dependent variable (y) decreases, resulting in a downward trend on the graph from left to right.
Example:
A car's fuel tank level showing a negative slope over time indicates fuel consumption.
Positive Slope
Indicates that as the independent variable (x) increases, the dependent variable (y) also increases, resulting in an upward trend on the graph from left to right.
Example:
A company's profit graph showing a positive slope indicates increasing profits over time.
Range
The set of all possible output values (y-values) that a function or relation can produce, corresponding to its defined domain.
Example:
For the function y = x², the range is y ≥ 0 since squaring any real number results in a non-negative value.
Slope (Δy/Δx)
A measure of the steepness and direction of a line or curve, calculated as the ratio of the change in y (vertical change) to the change in x (horizontal change) between two points.
Example:
A road with a slope of 0.05 means that for every 100 feet traveled horizontally, the elevation increases by 5 feet.
Solving for One Variable
The process of algebraically rearranging an equation to express one variable in terms of the other, which can reveal explicit functions from an implicit relation.
Example:
When solving for one variable in x = y², we get y = ±√x, showing two explicit functions.
Vertical Interval
A segment of a graph where the x-value remains constant while the y-value changes, meaning the rate of change of y with respect to x (Δy/Δx) is undefined, and the slope is undefined.
Example:
A wall in a video game, where the x-coordinate is fixed but the y-coordinate can vary, represents a vertical interval.