Graph of the Function Represented Analytically

Suppose we are given function that is represented analytically by the formula y=f(x).

Then its graph is a set of all points (x;y), where y=f(x), and x takes all values from the domain of f.

Example 1. Draw graph of y=x.graph of the function y=x

Graph of the function y=x is set of all points of the form (x;x), i.e. points that have same coordinates.

This set of points is bisector of first and third coordinate angles.

Example 2. Draw graph of the function `y=x^2`.

Let's write out a couple of values of function in table:

x -2 -1 -0.5 0 0.5 1 2
y `(-2)^2=4` `(-1)^2=1` `(-0.5)^2=0.25` `0^2=0` `(0.5)^2=0.25` `1^2=1` `2^2=4`

graph of the function y=x^2

Now draw points (0;0), (0.5;0.25), (-0.5;0.25), (1;1), (-1;1), (2;4), (-2;4) on coordinate plane and connect points with smooth line. We will obtain graph of the function `y=x^2`. This graph is called parabola. In general, parabola is graph of any function of the form `y=ax^2`, where `a!=0`.