# Function Graph

An example of a function graph

## How to Draw a Function Graph

First, start with a blank graph like this. It has x-values going left-to-right, and y-values going bottom-to-top:

The x-axis and y-axis cross over
where x and y are both zero.

## Plotting Points

A simple (but not perfect) approach is to calculate the function at some points and then plot them.

A function graph is the set of points of the values taken by the function.

### Example: y = x2 − 5

Let us calculate some points:

x y = x2−5
−2 −1
0 −5
1 −4
3 4

And plot them like this:

Looking better!

We can now guess that plotting all the points will look like this:

A nice parabola.

We should try to plot enough points to be confident in what is going on!

### Example: y = x3 − 5x

With these calculated points:

x y = x3−5x
−2 2
0 0
2 −2

We might think this is the graph:

But this is the real graph:

So "plotting some points" is useful, but can lead to mistakes.

## Complete Graph

For a graph to be "complete" we need to show all the important features:

• Crossing points
• Peaks
• Valleys
• Flat areas
• Asymptotes
• Any other special features

This often means thinking carefully about the function.

### Example: (x−1)/(x2−9)

On the page Rational Expressions we do some work to discover that the function:

• crosses the x-axis at 1,
• crosses the y-axis at 1/9,
• has vertical asymptotes (where it heads towards minus/plus infinity) at −3 and +3

The result is that we can make this sketch:

Sketch of (x−1)/(x2−9) from Rational Expressions.

We can also use Calculus (Finding Maxima and Minima using Derivatives)
to find some important features:

• The Function Grapher can help you. Enter the equation as "y=(some function of x)". You can use zoom to find important points.
• If you can't write the equation as "y=(some function of x)", you can try the Equation Grapher, where you enter equations like "x^2+y^2=9" (meaning x2+y2=9).

But remember they are just a help! They are only computer programs, and could easily miss some important thing on the graph, or not plot something correctly.

Note: you may hear the phrase "satisfy the equation", which means where the equation is true.