Use the table to answer the question

Question

x:8,9,10,11,12,....
y:29,32,35,38,41,...

Which of the following statements accurately describes the selection of the best model for the data?
1)A linear function would best fit this dataset, since it has constant 1st differences.
2)A quadratic function would best fit this dataset, since it has constant 1st differences.
3)A linear function would best fit this dataset, since it has constant 2nd differences.
4)A quadratic function would best fit this dataset, since it has constant 2nd differences.

Answer 1

To analyze the provided data, let's look at the values:

For \( x = 8, 9, 10, 11, 12 \):

• \( y = 29, 32, 35, 38, 41 \)

Now, we can find the differences in \( y \):

1st differences:

• From \( 29 \) to \( 32 \): \( 32 - 29 = 3 \)
• From \( 32 \) to \( 35 \): \( 35 - 32 = 3 \)
• From \( 35 \) to \( 38 \): \( 38 - 35 = 3 \)
• From \( 38 \) to \( 41 \): \( 41 - 38 = 3 \)

The first differences are constant (3), thus suggesting a linear relationship.

Now let's check the 2nd differences (which are the differences of the 1st differences):

• Since all the 1st differences are the same (3), the 2nd differences would be \( 0 \) (as \( 3 - 3 = 0 \)).

Since the first differences are constant, this confirms that the data is linear.

Based on the analysis, the correct statement is:

1. A linear function would best fit this dataset, since it has constant 1st differences.