A function can be graphically represented by the use of rectangular coordinates. If we represent the independent variable by $x$ and the corresponding value of the function by $y$, we can stick them together in an ordered pair $(x,y)$. This ordered pair will determine a point in the plane, and a number of such points will outline a curve indicating the correspondence of values of the independent and dependent variables. This curve is called the graph of the function.
By carefully drawing the graph of a function, a good idea is obtained of the behavior of the function as the independent variable changes. For example, the graph of the function $y=\log_{2}x$ is drawn in Figure 1. Here we see the following facts clearly pictured to the eye:
(a) For $x=1$, the value of $\log_{2}x$ is zero; that is, $\log_{2}1=0$.
(b) For $x>1$, $\log_{2}x$ is positive and increases as $x$ increases.
(c) For $0<x<1$, $\log_{2}x$ is negative and increases indefinitely in numerical value as $x$ diminishes.
(d) $\log_{2}x$ is not defined for $x=0$ because the logarithm of zero cannot be calculated.
To plot the graph of a function over an interval:
- construct a table of values by substituting several values for $x$ in the formula of the function $y=f(x)$;
- plot the $(x,y)$ points whose coordinates appear in the table of values;
- draw a smooth line free-hand through these points.
For example, suppose we want to plot the graph of $f(x)=x^{3}/2$ when $x\in[-2,2]$. First, we construct a table like the one below:
Now we can plot these points in the plane, and connect these points with a curve. As we will learn later, the curve connecting these points should be a smooth curve, but for now let’s simply connect them with straight lines (Figure 2 (a) ). For a finer representation of the graph, we can increase the number of $(x,y)$ points:
Now if we show these points in the plane and connect them, we will get Figure 2 (b).
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| (a) Graph of $y=x^3/2$ when connecting a few point with straight lines | (b) Graph of the same function when the number of points increases |
Figure 2
This is the way that we can plot functions over a (finite) interval using computer programs, such as MATLAB and Matplotlib (a plotting library for the Python programming language). First, we assume many values of $x$ and calculate the corresponding values of $y$; these programs can then plot this series of points and connect them with straight lines.