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Which of the following best models the data in the table?

Type of Function

The table presents values of x and y, showing how y increases as x increases
x y
2
8.5
4
46
6
107.5
8
193
(1 point)
Responses

linear
linear

cubic
cubic

exponential
exponential

quadratic

1 answer

To determine which type of function best models the provided data, we can analyze how the values of \( y \) change as \( x \) increases. Let's look at the given values:

  • \( x = 2 \), \( y = 8.5 \)
  • \( x = 4 \), \( y = 46 \)
  • \( x = 6 \), \( y = 107.5 \)
  • \( x = 8 \), \( y = 193 \)

  1. Calculate the first differences of \( y \):

    • From \( 2 \) to \( 4 \): \( 46 - 8.5 = 37.5 \)
    • From \( 4 \) to \( 6 \): \( 107.5 - 46 = 61.5 \)
    • From \( 6 \) to \( 8 \): \( 193 - 107.5 = 85.5 \)

    First differences: \( 37.5, 61.5, 85.5 \)

  2. Calculate the second differences:

    • From \( 37.5 \) to \( 61.5 \): \( 61.5 - 37.5 = 24 \)
    • From \( 61.5 \) to \( 85.5 \): \( 85.5 - 61.5 = 24 \)

    The second differences are constant, indicating a quadratic function.

Since we can see that the second differences are constant, the best model for this data is:

Quadratic.