Ch. 3 Linear Functions Vocabulary Terms

Continuous Domain

A continuous domain is a set of input values that consists of all numbers in an interval. Example: All numbers from 1 to 5

Discrete Domain

A discrete domain is a set of input values that consists of only certain numbers in an interval. Example: Integers from 1 to 5

Family of Functions

A family of functions is a group of functions with similar characteristics.

Linear Equation in Two Variables

A linear equation in two variables, x and y, is an equation that can be written in the form y = mx + b, where m and b are constants. The graph of a linear equation is a line.

Constant Function

A linear equation written in the form y = 0x + b, or y = b, is a constant function. The graph of a constant function is a horizontal line.

Slope-Intercept Form

A linear equation written in the form y = mx + b is written in slope-intercept form. The slope of the line is m, and the y-intercept of the line is b.

Linear Function

A linear function is a function whose graph is a nonvertical line. A linear function has a constant rate of change and can be represented by a linear equation in two variables.

Nonlinear Function

A nonlinear function does not have a constant rate of change. So, its graph is not a line.

Relation

A relation pairs inputs with outputs. When a relation is given as ordered pairs, the x-coordinates are inputs and the y-coordinates are outputs.

Function

A relation that pairs each input with exactly one output is a function.

Solution of an Equation in Two Variables

A solution of a linear equation in two variables is an ordered pair (x,y) that makes the equation true. The graph of a linear equation in two variables is the set of points (x,y) in a coordinate plane that represents all solutions of the equation. Sometimes the points are distinct, and other times the points are connected.

Reflection

A transformation that flips a graph over a line called the line of reflection. Reflections in the x-axis: The graph of y = -f(x) is a reflection in the x-axis of the graph of y = f(x). Multiplying the outputs by -1 changes their signs. Reflections in the y-axis: The graph of y = f(-x) is a reflection in the y-axis of the graph of y = f(x). Multiplying the inputs by -1 changes their signs.

Translation

A transformation that shifts a graph horizontally or vertically but does not change the size, shape, or orientation of the graph. Horizontal translations: The graph of y = f(x - h) is a horizontal translation of the graph of y = f(x), where h is not 0. Subtracting h from the inputs before evaluating the function shifts the graph left when h < 0 and right when h > 0. Vertical translations: The graph of y = f(x) + k is a vertical translation of the graph of y = f(x), where k is not 0. Adding k to the outputs shifts the graph down when k < 0 and up when k > 0.

Transformation

Changes the size, shape, position, or orientation of a graph

Slope

Slope is the rate of change between any two points on a line. It is the measure of the steepness of the line. It is referred to as m in the equation y = mx + b. The slope m of a nonvertical line passing through the points (x1, y1) and (x2, y2) is the ratio of the rise (change in y) to the run (change in x). slope = m = rise/run = (y2 - y1)/(x2 - x1)

Run

The change in x

Rise

The change in y

Domain

The domain of a function is the set of all possible input values.

Equations of Horizontal Lines

The equation of a horizontal line can be written in the form y = b. The line passes through the point (0, b).

Equations of Vertical Lines

The equation of a vertical line can be written in the form x = a. The line passes through the point (a, 0). A vertical line is not a function.

Parent Function

The most basic function in a family of functions. For nonconstant linear functions, the parent function is f(x)=x. The graphs of all other nonconstant linear functions are transformations of the graph of the parent function.

Function Notation

The notation, f(x), called function notation, is another name for y. The notation is read as "the value of f at x" or "f of x." The parentheses do not imply multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions.

Range

The range of a function is the set of all possible output values.

Standard Form

The standard form of a linear equation is Ax + By = C, where a, b, and c are real numbers and A and B are not both zero.

Independent Variable

The variable that represents the input values of a function is the independent variable because it can be any value in the domain.

Dependent Variable

The variable that represents the output values of a function is the dependent variable because it depends on the value of the independent variable. When an equation represents a function, the dependent variable is defined in terms of the independent variable. The statement "y is a function of x" means that y varies depending on the value of x.

x-intercept

The x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. The y-coordinate of the x-intercept is always 0.

y-intercept

The y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis. The x-coordinate of the y-intercept is always 0.

Negative Slope

m < 0 and the graph of the line falls from left to right

Slope of 0

m = 0 and the graph of the line is horizontal

Undefined Slope

m = undefined and the graph of the line is vertical

Positive Slope

m >0 and the graph of the line rises from left to right

Input Values

x values, independent variable values, domain values

Output Values

y values, dependent variable values, range values