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7.3 Quadratic Patterns

7.3 Quadratic Patterns
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  • 19. Revenue

    Data from real-world applications rarely match a linear, exponential, or quadratic model perfectly. The table shows the revenue for selling various units. Determine whether a linear, exponential, or quadratic model best represents the data in the table. Explain your reasoning.

    data folder

    • Worked-Out Solution

      Use a spreadsheet to analyze the data. The first differences are about equal. So, a linear model best represents the data.

      data folder

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      ____      ______    _  _    __   __    ______  
     |  _ \\   /_   _//  | \| ||  \ \\/ //  /_   _// 
     | |_| ||   -| ||-   |  ' ||   \ ` //     | ||   
     | .  //    _| ||_   | .  ||    | ||     _| ||   
     |_|\_\\   /_____//  |_|\_||    |_||    /__//    
     `-` --`   `-----`   `-` -`     `-`'    `--`     
                                                     
    
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  • 20. Cost of Production

    Data from real-world applications rarely match a linear, exponential, or quadratic model perfectly. The table shows the total cost for producing various units. Determine whether a linear, exponential, or quadratic model best represents the data in the table. Explain your reasoning.

    data folder

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

    Data from real-world applications rarely match a linear, exponential, or quadratic model perfectly. The table shows the profit from selling various units. Determine whether a linear, exponential, or quadratic model best represents the data in the table. Explain your reasoning.

    data folder

    • Worked-Out Solution

      Use a spreadsheet to analyze the data. The second differences are about equal. So, a quadratic model best represents the data.

      data folder

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     __   _      ___      ______   __   __    ______  
    | || | ||   / _ \\   /_   _//  \ \\/ //  /_____// 
    | '--' ||  / //\ \\  `-| |,-    \ ` //   `____ `  
    | .--. || |  ___  ||   | ||      | ||    /___//   
    |_|| |_|| |_||  |_||   |_||      |_||    `__ `    
    `-`  `-`  `-`   `-`    `-`'      `-`'    /_//     
                                             `-`      
    
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  • 22. Stock Price

    Data from real-world applications rarely match a linear, exponential, or quadratic model perfectly. The table shows the stock price of a company for various years. Determine whether a linear, exponential, or quadratic model best represents the data in the table. Explain your reasoning.

    data folder

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      ______    ______  __    __    _____     _____   
     /_   _//  /_   _// \ \\ / //  |  ___||  /  ___|| 
       | ||     -| ||-   \ \/ //   | ||__   | // __   
      _| ||     _| ||_    \  //    | ||__   | \\_\ || 
     /__//     /_____//    \//     |_____||  \____//  
     `--`      `-----`      `      `-----`    `---`   
                                                      
    
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  • 23. Completing a Table

    Fold a rectangular piece of paper in half. Open the paper and record the number of folds and the number of sections created. Repeat this process four times and increase the number of folds by one each time.

    2 Folds
    4 Sections

    Complete the table.

    • Worked-Out Solution

      The completed table is shown below.

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      _____     ______   _____       ___      ______  
     /  ___||  /_   _// |  __ \\    / _ \\   /_   _// 
    | // __     -| ||-  | |  \ ||  | / \ ||    | ||   
    | \\_\ ||   _| ||_  | |__/ ||  | \_/ ||   _| ||   
     \____//   /_____// |_____//    \___//   /__//    
      `---`    `-----`   -----`     `---`    `--`     
                                                      
    
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  • 24. Graphing Data

    Graph the data in Exercise 23. Determine whether the pattern is linear, exponential, or quadratic.

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      ______   _    _     _  __    ______   _    _   
     /_   _// | || | ||  | |/ //  /_   _// | \  / || 
     `-| |,-  | || | ||  | ' //    -| ||-  |  \/  || 
       | ||   | \\_/ ||  | . \\    _| ||_  | .  . || 
       |_||    \____//   |_|\_\\  /_____// |_|\/|_|| 
       `-`'     `---`    `-` --`  `-----`  `-`  `-`  
                                                     
    
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  • 25. Writing a Formula

    Fold a rectangular piece of paper in half. Open the paper and record the number of folds and the number of sections created. Repeat this process four times and increase the number of folds by one each time.

    2 Folds
    4 Sections

    Write a formula for the model that represents the data.

    • Worked-Out Solution

      Each time the paper is folded, the number of sections doubles.

      Let n be the number of folds. Let S be the number of sections.

      A formula that relates n and S is

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    __    __     ___      _____      ___      _____   
    \ \\ / //   / _ \\   |__  //    / _ \\   /  ___|| 
     \ \/ //   / //\ \\    / //    | / \ || | // __   
      \  //   |  ___  ||  / //__   | \_/ || | \\_\ || 
       \//    |_||  |_|| /_____||   \___//   \____//  
        `     `-`   `-`  `-----`    `---`     `---`   
                                                      
    
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  • 26. Folding Paper

    Fold a rectangular piece of paper in half. Open the paper and record the number of folds and the number of sections created. Repeat this process four times and increase the number of folds by one each time. How many sections are created after eight folds?

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