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7.3 Quadratic Patterns

7.3 Quadratic Patterns
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  • 13. Gravity on the Moon

    An object is dropped from a height of 50 feet on the moon. The table shows the distances it has fallen at various times.

    data folder

    Determine whether the pattern in the table is linear, exponential, quadratic, or none of these. Explain your reasoning. (See Example 5 and Example 6.)

    • 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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  • 14. Gravity on Venus

    An object is dropped from a height of 150 feet on Venus. The table shows the distances it has fallen at various times.

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    Determine whether the pattern in the table is linear, exponential, quadratic, or none of these. Explain your reasoning. (See Example 5 and Example 6.)

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     _____     _    _    _    _    _    _   __    __  
    |  __ \\  | || | || | |  | || | || | || \ \\ / // 
    | |  \ || | || | || | |/\| || | || | ||  \ \/ //  
    | |__/ || | \\_/ || |  /\  || | \\_/ ||   \  //   
    |_____//   \____//  |_// \_||  \____//     \//    
     -----`     `---`   `-`   `-`   `---`       `     
                                                      
    
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  • 15. Gravity on Mars

    An object is dropped from a height of 300 feet on Mars. The table shows the distances it has fallen at various times.

    data folder

    Determine whether the pattern in the table is linear, exponential, quadratic, or none of these. Explain your reasoning. (See Example 5 and Example 6.)

    • 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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  • 16. Gravity on Jupiter

    An object is dropped from a height of 1600 feet on Jupiter. The table shows the heights of the object at various times.

    data folder

    Determine whether the pattern in the table is linear, exponential, quadratic, or none of these. Explain your reasoning. (See Example 5 and Example 6.)

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      ______     ___               _    _     ______  
     /_   _//   / _ \\      ___   | || | ||  /_____// 
     `-| |,-   / //\ \\    /   || | || | ||  `____ `  
       | ||   |  ___  ||  | [] || | \\_/ ||  /___//   
       |_||   |_||  |_||   \__ ||  \____//   `__ `    
       `-`'   `-`   `-`     -|_||   `---`    /_//     
                             `-`             `-`      
    
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  • 17. Sign of the Second Differences

    Graph the data in Exercises 14 and 15 on the same coordinate plane. Compare the graphs. What appears to be the relationship between the sign of the second differences and the corresponding graph?

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    Table from Exercise 14: Object dropped on Venus

     


    Table from Exercise 15: Object dropped on Mars

    • Worked-Out Solution

      Use a spreadsheet to analyze the data.

      For Exercise 14, the data values are increasing and the second differences are positive.

      For Exercise 15, the data values are decreasing and the second differences are negative.

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                ______    _____     _____    ______   
        ___    /_   _//  / ____||  |  ___|| |      \\ 
       /   ||   -| ||-  / //---`'  | ||__   |  --  // 
      | [] ||   _| ||_  \ \\___    | ||__   |  --  \\ 
       \__ ||  /_____//  \_____||  |_____|| |______// 
        -|_||  `-----`    `----`   `-----`  `------`  
         `-`                                          
    
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  • 18. Comparing Gravitational Forces

    The moon's gravitational force is much less than that of Earth. Use the table in Exercise 13 and the table in Example 5 to estimate how many times stronger Earth's gravitational force is than the moon's gravitational force. Explain your reasoning.

    Table from Example 5: Object dropped on Earth

     

    Table from Exercise 13: Object dropped on the moon

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