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7.3 Quadratic Patterns

7.3 Quadratic Patterns
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  • 1. Height of a Football

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

    data folder

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

    • Worked-Out Solution

      Use a spreadsheet to analyze the data.

      data folder

      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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      _____     _____     _  _      ___      _  __  
     |__  //   |  ___||  | \| ||   / _ \\   | |/ // 
       / //    | ||__    |  ' ||  | / \ ||  | ' //  
      / //__   | ||__    | .  ||  | \_/ ||  | . \\  
     /_____||  |_____||  |_|\_||   \___//   |_|\_\\ 
     `-----`   `-----`   `-` -`    `---`    `-` --` 
                                                    
    
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  • 2. Rushing Yards

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

    data folder

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

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

    data folder

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

    • Worked-Out Solution

      Use a spreadsheet to analyze the data.

      data folder

      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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    __    __     ___      ______   __   __            
    \ \\ / //   / _ \\   /_____//  \ \\/ //   ____    
     \ \/ //   | / \ ||  `____ `    \ ` //   |    \\  
      \  //    | \_/ ||  /___//      | ||    | [] ||  
       \//      \___//   `__ `       |_||    |  __//  
        `       `---`    /_//        `-`'    |_|`-`   
                         `-`                 `-`      
    
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  • 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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     _    _      ___               _    _             
    | || | ||   / _ \\      ___   | || | ||   ____    
    | || | ||  / //\ \\    /   || | || | ||  |    \\  
    | \\_/ || |  ___  ||  | [] || | \\_/ ||  | [] ||  
     \____//  |_||  |_||   \__ ||  \____//   |  __//  
      `---`   `-`   `-`     -|_||   `---`    |_|`-`   
                             `-`             `-`      
    
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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.)

    data folder

    • Worked-Out Solution

      Use a spreadsheet to analyze and graph the data.

      data folder

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