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• 1. Height of a Football

The table shows the heights of a football at various times after a punt.

Describe the pattern in the table. (See Example 1 and Example 2.)

• Worked-Out Solution

Use a spreadsheet to analyze the data.

The first differences are not equal, so the pattern is not linear.

The consecutive ratios are not equal, so the pattern is not exponential.

The second differences are equal. So, the pattern is quadratic.

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``` __   _     _____     _____      ___      ______
| || | ||  |  ___||  /  ___||   / _ \\   /_   _//
| '--' ||  | ||__   | // __    | / \ ||    | ||
| .--. ||  | ||__   | \\_\ ||  | \_/ ||   _| ||
|_|| |_||  |_____||  \____//    \___//   /__//
`-`  `-`   `-----`    `---`     `---`    `--`

```
• 2. Rushing Yards

The table shows the distances gained by a running back after various numbers of rushing attempts.

Describe the pattern in the table.  (See Example 1 and Example 2.)

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```            ______               ___      ______
___    /_   _//     ___     / _ \\   /_   _//
/   ||   -| ||-     /   ||  / //\ \\  `-| |,-
| [] ||   _| ||_    | [] || |  ___  ||   | ||
\__ ||  /_____//    \__ || |_||  |_||   |_||
-|_||  `-----`      -|_|| `-`   `-`    `-`'
`-`                 `-`
```
• 3. Height of a Football

The table shows the heights of a football at various times after a field goal attempt.

Describe the pattern in the table. (See Example 1 and Example 2.)

• Worked-Out Solution

Use a spreadsheet to analyze the data.

The first differences are not equal, so the pattern is not linear.

The consecutive ratios are not equal, so the pattern is not exponential.

The second differences are equal. So, the pattern is quadratic.

These comments are not screened before publication. Constructive debate about the information on this page is welcome, but personal attacks are not. Please do not post comments that are commercial in nature or that violate copyright. Comments that we regard as obscene, defamatory, or intended to incite violence will be removed. If you find a comment offensive, you may flag it.

```  ______     ___     _    _    _    _    __   _
/_____//   / _ \\  | \  / || | || | || | || | ||
`____ `   / //\ \\ |  \/  || | || | || | '--' ||
/___//   |  ___  ||| .  . || | \\_/ || | .--. ||
`__ `    |_||  |_|||_|\/|_||  \____//  |_|| |_||
/_//     `-`   `-` `-`  `-`    `---`   `-`  `-`
`-`
```
• 4. Punt Height

In Exercise 1, extend the pattern to find the height of the football after 4 seconds. (See Example 1.)

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```  _  _      ___     __   __    _____     _  _
| \| ||   / _ \\   \ \\/ //  |  ___||  | \| ||
|  ' ||  | / \ ||   \   //   | ||__    |  ' ||
| .  ||  | \_/ ||   / . \\   | ||__    | .  ||
|_|\_||   \___//   /_//\_\\  |_____||  |_|\_||
`-` -`    `---`    `-`  --`  `-----`   `-` -`

```
• 5. Passing a Football

The table shows the heights of a football at various times after a quarterback passes it to a receiver. Use a spreadsheet to graph the data. Describe the graph. (See Example 2.)

• Worked-Out Solution

Use a spreadsheet to analyze and graph the data.

The first differences are not equal, so the pattern is not linear.

The consecutive ratios are not equal, so the pattern is not exponential.

The second differences are equal. So, the pattern is quadratic.

The graph is a U-shaped curve that curves downward. This curve is parabolic.

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```  ____     _    _                ___     _    _
|  _ \\  | || | ||   ____      / _ \\  | \  / ||
| |_| || | || | ||  |    \\   | / \ || |  \/  ||
| .  //  | \\_/ ||  | [] ||   | \_/ || | .  . ||
|_|\_\\   \____//   |  __//    \___//  |_|\/|_||
`-` --`    `---`    |_|`-`     `---`   `-`  `-`
`-`
```
• 6. Interpreting a Graph

Use the graph in Exercise 5 to determine how long the height of the football increases.

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```           _    _     ______     ___      ______