Modeling Linear Relationships and Functions

Slope

Describes the steepness of the line. Is the change in y over the change in x. Is the rise over run.

Slope-Intercept Form

A way of writing an equation, so the slope and y-intercept are easily identifiable. y=mx+b

Function

A relation where each member of the domain is mapped to one and only member of the range.

Function (Graph)

If any vertical line crosses the graph in more than one place, then the graph is not a function. If each vertical line crosses the graph at exactly one point, the graph is a function.

Linear Function (Equation)

Any equation that can be written in slope intercept form.

Increasing Interval

A portion of a function or graph where the output values (y-vaules) get larger as the input values (x-values) increase.

Decreasing Interval

A portion of a function or a graph where the output values (y-values) get smaller as the input values (x-values) increase.

Constant Interval

A portion of a function or graph where the output values (y-values) stay the same as the input values (x-values) increase.

Initial Value

In a real-world application, it’s the starting amount or condition before any changes happen. It represents f(0), the output of the function when the input is zero.

Domain

All the x-values for which the function is defined. It is the set of input values that you can put into a function.

Range

All the y-values the function can reach. The set of all possible output values that mathematical relationship or equation can produce.

Vertical Line Test

A way to determine if a graph represents a function. By drawing a vertical line across the graph, if it touches the graph at more than one point at any location, then the graph is not a function. If it only touches at one point everywhere, then the graph is a function.

When is a function positive?

In the regions where the functions graph is above the x-axis. In these intervals, the y-values are greater than zero. f(x)<0

When is a Function Negative?

In the regions where the function graph is below the x-axis. In these intervals, the y-values are less than zero. f(x)>0