Graph of a function is a graphical representation of such function.

## Coordinate System

Before we draw our first graph of a function we need to know some basics. We need a *coordinate system*. It’s quite simple, we draw two straight lines intersecting at right angle. We will call one of the line *horizontal axis* and the second one *vertical axis*. See example:

It’s not much, so far. We can label both the axes by real numbers so we can measure distances there.

The intersection point is call *the origin*. Now we are able to define a *coordinates* for each point on the place. For example, let’s suppose a point *A* placed right there:

We can see that its *horizontal* coordinate is *1* and its *vertical* coordinate is *3*. We can draw a simple segments that helps us determine the coordinates. Now we can write that the coordinates of the point *A* are *(1, 3)*. Actually each pair *(a, b)* of real numbers are coordinates of a single point on such coordinate system. Let’s say we write down coordinates *B = ( − 2, − 1)*. We can easily find them: we find * − 2* on the horizontal axis and * − 1* on the vertical axis. Then the point *B* lays right there:

## Ordered Pair

In the previous chapter we used notation *(a, b)* for coordinates. We’ll call the *(a, b)* an *ordered pair*. What does it mean, *ordered*? Well it means that *(a, b)* is a different pair than the pair *(b, a)*. For example: the ordered pair *(2, 3)* differs from ordered pair *(3, 2*). We can see it on the graph:

We could say that two ordered pair *(a, b)* and *(c, d)* equals when

## How to Draw a Graph of a Function

A function has defined certain outputs for its inputs. For example, for a function *f(x) = 2x* we have for input *x = 3* the output *f(3) = 6*. For an input *x = 1* we have the output *f(1) = 2*. We can see all *inputs* as a first coordinate and all *outputs* as a second coordinate of an ordered pair.

Basically we can say that the function is set of ordered pairs in the form *(x, f(x))* and in this way we can draw the graph of the function. So we know that *f(3) = 6*. It means that the point *(3, 6)* is part of the graph of the function. If we want to draw the whole graph we just need to draw *all* such points. We can start from the two points we already know, *(1, 2)* and *(3, 6)*:

These two points are part of the graph. We can add a few more points, just a few examples:

- Because
*f(0) = 0*, then*(0, 0)*is part of the graph, *f(0.5) = 1*, then*(0.5, 1)*is part of the graph,*f(2) = 4*, then*(2, 4)*is part of the graph.- …

We can add those three points to the graph:

We can kind of see the pattern: all the points lays on the same line. In the end we can draw the graph of the function *f(x) = 2x* as a simple line:

This red line represents all the points in the form *(x, f(x))* for all *x*.

## More Examples of Function Graphs

Let’s take a function *f(x) = x + 1*. We can compute some of its values:

*f(0) = 1*thus we get coordinates*(0, 1)*,*f(2) = 3*thus we get coordinates*(2, 3)*,*f(3) = 4*thus we get coordinates*(3, 4)*,*f( − 2) = − 1*thus we get coordinates*( − 2, − 1)*,- …

Now we’ll draw these points:

We can see that all point are on the same line, again. The whole graph looks like this:

But not all graphs are formed by a straight line. Let’s take another example: *f(x) = x ^{2}*.

*f(0) = 0*thus we get coordinates*(0, 0)*,*f(1) = 1*thus we get coordinates*(1, 1)*,*f(2) = 4*thus we get coordinates*(2, 4)*,*f(3) = 9*thus we get coordinates*(3, 9)*,*f( − 1) = 1*thus we get coordinates*( − 1, 1)*,*f( − 2) = 4*thus we get coordinates*( − 2, 4)*,*f( − 3) = 9*thus we get coordinates*( − 3, 9)*,- …

We can clearly see that these points doesn’t form any line. If we add all remaining point of function *f* we’ll get this graph:

This curve is also known as *parabola*.

## What Can We Read From a Graph

We can go the other way around and we determine the function definition from a graph. Let’s take this graph:

What is the output for the input *x = 1*? We can determine it by looking at the graph. We found *x = 1* on the horizontal axis and then we’ll find appropriate coordinate on the vertical axis that gives us the full coordinates:

The graph includes point *(1, 3)* which means that *f(1) = 3*. We can see that the graph contains also these points:

*(0, 2)*meaning*f(0) = 2*,*( − 1, 1)*meaning*f( − 1) = 1*,*( − 2, 0)*meaning*f( − 2) = 0*,- …

The graph clearly expresses the function *f(x) = x + 2*.

## Is Every Curve a Graph of Some Function?

What do you think? Is this a graph of some function?

What is the output for input *x = 1*? I. e. what is *f(1)*? Let’s take a look in the graph:

Clearly, we can see that there are two *output* for a single *input*. The outputs are denoted by the points *A* and *B*. This graph cannot represent a graph of a function since there are more outputs for a single input which violates the most fundamental part of being a function.

Remember: a curve in the plane represents a function if and only if any vertical line intersects the graph in at most one point. In can intersect the graph in *zero points*, meaning it doesn’t intersect the graph at all, but it cannot intersect the graph at two or more points.