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  Theory of Graphs of Functions

Basic Definitions of Graphs of Functions:

Graph of a function: Graph of a function f is the set of all ordered pairs (x, f(x)); here x is in the domain of f.

Increasing Function: It is a function which is increasing on an open interval when the function increases (that is, positive slope) on an interval as you move from left to right.

Decreasing Function: It is a function which is decreasing on an open interval when the function drops (that is, negative slope) on an interval as you move from left to right.

Constant Function: A function is constant on an open interval when the function remains constant (that is, horizontal line segment) on an interval as you move from left to right.

Relative Minimum: The function consists of a relative minimum at x = a when the function computed at x=a is less than at any other point in the neighborhood surrounding x = a. The relative minimum is the lowest point in an open interval, however not necessarily over the whole domain. The relative minimums take place if the function is reducing to the left of the point and rising to the right of point.

Relative Maximum: The function consists of a relative maximum at x = a when the function computed at x = a is bigger than at any other point in the neighborhood surrounding x = a. The relative maximum is the maximum point in an open interval, however not necessarily over the whole domain. The Relative maximums take place if the function is rising to the left of the point and reducing to the right of point.

Greatest Integer Function: The greatest integer of a value is the maximum integer less than or equivalent to the value.

Symmetry about the y-axis: It is a relation in symmetric regarding y-axis if for each and every point (x, y) on the graph, the point (-x, y) is as well on the graph.

Symmetry about the x-axis: It is a relation in symmetric regarding the x-axis if for each and every point (x, y) on graph, the point (x,-y) is as well on the graph.

Symmetry about the origin: It is a relation is symmetric regarding the origin if for each and every point (x,y) on the graph, the point (-x,-y) is as well on the graph.

Even Function: The function is even if for each x in the domain of the function, f(-x) = f(x)

Odd Function: The function is odd if for each x in the domain of the function, f(-x) = -f(x)

Vertical Line Test:

The relation is a function when all vertical lines drawn via the graph of the relation intersect in no more than one point.

The contra positive of that is frequently used.

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

Greatest Integer Function:

The greatest integer function is frequently termed as the Integer function, and is abbreviated INT on the calculator. You might find the INT function on the calculator by going to the [Math] menu, arrowing right to the NUM option, and then selecting the INT function (it is number 5 on TI83).

The Integer function is at times termed as the step function as of the stair step result obtained when graphing it. Be sure to employ a decimal setting whenever graphing the maximum integer function or you will get weird outcomes. You might also wish to use Dot mode rather than Connected mode whenever graphing the Integer function. You can modify modes on the TI series by pressing the [Mode] key.

Mathematically, the maximum integer function is symbolized by using a double left bracket and double right bracket.

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Symmetry-Odd/Even Functions:

Symmetry about y-axis:

Symmetry about the y-axis signifies that the left side of graph is a mirror image of the right side of graph. Mathematically, a relation that is symmetric regarding y-axis comprises the property that for each and every point (x,y) which is on graph, the point (-x,y) is as well on the graph. In another word, to mirror something about the y-axis, take the opposite of all x-coordinates and leave the y-coordinates only.

Symmetry about x-axis:

Symmetry regarding x-axis signifies that the bottom side of the graph is a mirror image of top side of graph. Mathematically, the relation that is symmetric regarding x-axis consists of the property that for each and every point (x, y) which is on the graph, the point (x,-y) is also on the graph. In another word, to mirror something about the x-axis, take the opposite of all y-coordinates and leave the x-coordinates only.

Symmetry about the origin:

The Symmetry about the origin signifies that for each and every point (x, y) on the graph, the point (-x,-y) is as well on the graph. Graphically, to make symmetry about the origin, take any point, draw an imaginary line via the origin, and put a point on that line at similar distance as the original point was from origin on the other side of origin.

Even Function:

The function is even when it is symmetric about the y-axis. Mathematically, as the y-coordinates (that is, the values of function) have to be equivalent, and the x-coordinates are opposite, one can write: f(-x) = f(x).

There is a reason such functions are termed as even. When you have a polynomial function in one variable, all the exponents on independent variable will be even. Keep in mind that a constant is zero power the variable and zero is even.

Odd Function:

The function is odd when it is symmetric about the origin. Mathematically, as both x-coordinate and y-coordinates are negated, one can write: f(-x) = -f(x).

There is a reason such functions are termed as odd. When you have a polynomial function in one variable, then all the exponents on independent variable will be odd. Keep in mind that a constant is zero power of the variable and will be even; therefore if there is a constant term, then it is not an odd function.

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