The Nature of Graphs
Introduction to Graphs
W E WILL BEGIN with some basic vocabulary.
First, a coordinate . A coordinate is a number. It labels a point on a line.
The coordinates 0, 1, 2, 3, etc. label those points. They are the "addresses" of those points.
A coordinate axis is a line with coordinates.
To label the points in a plane, we will need more than one coordinate axis, and we place them at right angles. Hence, they are called rectangular coordinate axes . And the coordinates on them are called rectangular coordinates . They are also called Cartesian coordinates ,
Finally, the rectangular coordinates of a point are an ordered pair , ( x , y ). (2, 3) labels a different point than (3, 2). The x -coordinate is always entered first, and the y -coordinate second.
BASIC GRAPHS
A constant function:
Here is the graph of y = f ( x ) = 3. It is a straight line parallel to the x -axis. It is called a constant function , because to every value of x there corresponds the same value of y : 3.
A constant function has the form
y = c ,
where c is a constant, that is, a number.
The identity function and the absolute value function:
y = x is called the identity function, because the value of y is identical with that of x . The coordinate pairs are ( x , x ).
In the absolute value function, the negative values of y in the identity function are reflected into the positive side. For, |− x | = | x | = x . The coördinate pairs are ( x , | x |).
Example.
a) What is the domain of the identity function?
There is no natural restriction on the values of x . Therefore, the domain -- where the function "lives" -- includes every real number.
−  < x < 
b) What is the range of the identity function?
The range are those values of y that correspond to the values in the domain. Inspecting the graph will show that y , also, will take every real value.
−  < y < 
Parabola and square root function:
In the parabola y = x ², the coördinate pairs are ( x , x ²). We can see that the following points are on the graph: (1, 1), (−1, 1), (2, 4), (−2, 4), and so on.
The graph of the square root function is related to y = x ². It is its inverse . The coordinate pairs are ( x ,  ). For example, (1, 1), (4, 2), (9, 3), and so on.
Notice that the square root function is defined only for non-negative values of x . For, the square root of a negative number is not real.
The cubic function:
The cubic function is y = x ³. When x is negative, y is negative -- odd powers of a negative number are negative.
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