# 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 = x^{2} − 5

Let us calculate **some points**:

x | y = x^{2}−5 |
---|---|

−2 | −1 |

0 | −5 |

1 | −4 |

3 | 4 |

And plot them like this:

Not very helpful yet. Let us add some **more points**:

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 = x^{3} − 5x

With these calculated points:

x | y = x^{3}−5x |
---|---|

−2 | 2 |

0 | 0 |

2 | −2 |

We might think this is the graph:

But this is the real graph:

**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)/(x^{2}−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)/(x^{2}−9) from Rational Expressions.

## Using Calculus

We can also find Maxima and Minima using derivatives :

## Tools to Help You

- 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
**x**).^{2}+y^{2}=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**.