When plotting points to graph a function, it is important to plot enough points to represent the important information about the function. After plotting several points, fill in missing information as needed by plotting additional points. It is also helpful to have a general idea of the shape of the graph based on the type of function.

Make a table of values by evaluating the function for several values of $x$.

$x$ |
$f(x)=x^{2}+3$ |
$y=f(x)$ |
---|---|---|

$-2$ |
$(-2)^2+3$ |
$7$ |

$-1$ |
$(-1)^2+3$ |
$4$ |

$0$ |
$0^2+3$ |
$3$ |

$1$ |
$1^2+3$ |
$4$ |

$2$ |
$2^2+3$ |
$7$ |

Use the table to identify the points on the graph.

$x$ |
$y=f(x)$ |
$(x, y)$ |
---|---|---|

$-2$ |
$7$ |
$(-2,7)$ |

$-1$ |
$4$ |
$(-1,4)$ |

$0$ |
$3$ |
$(0,3)$ |

$1$ |
$4$ |
$(1,4)$ |

$2$ |
$7$ |
$(2,7)$ |

### Vertical Line Test

Any two distinct points on a vertical line have the same $x$-coordinate and different $y$-coordinates. So if two or more points in a relation are on the same vertical line, then the $x$-value corresponds to multiple $y$-values. Using vertical lines on a graph to determine whether a relation is a function is called the **vertical line test**. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

To determine whether the graph of a relation represents a function:

1. Draw or imagine vertical lines running through the graph.

2. Look for any places that a vertical line would cross or touch the graph more than one time. These represent an $x$-value that corresponds to more than one $y$-value.

- If every vertical line crosses the graph only once, the relation is a function.
- If any vertical line crosses the graph more than once, the relation is not a function.

Function | Not a Function |
---|---|

The graph represents the relation $y=x^2$. There is no vertical line that touches the graph more than once. The relation is a function. | The graph of the relation is in the shape of a circle. Each vertical line intersects the circle at two different $y$-values. Therefore, the relation is not a function. |

### Piecewise Functions

**piecewise function**has different rules applied to different intervals in the domain. To graph a piecewise function, graph the pieces separately over the intervals for which they are defined.

Analyze the first piece of the function.

When the $x$-values are less than zero, this rule applies:Analyze the second piece of the graph.

When the $x$-values are greater than or equal to zero, this equation applies:Graph the pieces together.

The first piece is: