# 1.4 - Graphs of Functions

## Definitions

These definitions are mathematically loose (that means a mathematician would pull his or her hair out but a normal person might understand them).

Graph of a function
The graph of a function f is the set of all ordered pairs ( x, f(x) ) where x is in the domain of f.
Increasing Function
A function is increasing on an open interval if the function rises (positive slope) on the interval as you move from left to right.
Decreasing Function
A function is decreasing on an open interval if the function falls (negative slope) on the interval as you move from left to right.
Constant Function
A function is constant on an open interval if the function remains constant (horizontal line segment) on the interval as you move from left to right.
Relative Minimum
A function has a relative minimum at x=a if the function evaluated at x=a is less than at any other point in the neighborhood surrounding x=a. A relative minimum is the lowest point in an open interval, but not necessarily over the entire domain. Relative minimums occur when the function is decreasing to the left of the point and increasing to the right of the point.
Relative Maximum
A function has a relative maximum at x=a if the function evaluated at x=a is greater than at any other point in the neighborhood surrounding x=a. A relative maximum is the highest point in an open interval, but not necessarily over the entire domain. Relative maximums occur when the function is increasing to the left of the point and decreasing to the right of the point.
Greatest Integer Function
The greatest integer of a value is the largest integer less than or equal to the value.
A relation is symmetric about the y-axis if for every point (x,y) on the graph, the point (-x,y) is also on the graph.
A relation is symmetric about the x-axis if for every point (x,y) on the graph, the point (x,-y) is also on the graph.
A relation is symmetric about the origin if for every point (x,y) on the graph, the point (-x,-y) is also on the graph.
Even Function
A function is even if for every x in the domain of the function, f(-x) = f(x)
Odd Function
A function is odd if for every x in the domain of the function, f(-x) = -f(x)

## Vertical Line Test

A relation is a function if all vertical lines drawn through the graph of the relation intersect in no more than one point.

The contrapositive of that is often used.

If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function.

## Greatest Integer Function

The greatest integer function is often called the Integer function (or Floor in upper level mathematics), and is abbreviated INT on the calculator. You may find the INT function on the calculator by going into the [Math] menu, arrowing right to the NUM option, and then choosing the INT function (it's number 5 on the TI83).

The Integer function is sometimes called the step function because of the stair step effect obtained when graphing it. Be sure to use a decimal setting when graphing the greatest integer function, or you will get weird results. You may also want to use Dot mode instead of Connected mode when graphing the Integer function. You can change modes on the TI series by hitting the [Mode] key.

Mathematically, the greatest integer function is represented using a double left bracket and double right bracket