Question

This table shows the relationship between x and y x y 2 2 5 -2 8 -6 Which type of function best models the data in the table? (1 point) Responses exponential exponential cubic cubic linear linear quadratic

Answer 1

To determine which type of function best models the relationship between \( x \) and \( y \) from the given data points:

\[
\begin{align*}
(2, 2) \\
(5, -2) \\
(8, -6) \\
\end{align*}
\]

Let's analyze the changes in \( y \) as \( x \) changes:

1. From \( x = 2 \) to \( x = 5 \):
- Change in \( y \): \( -2 - 2 = -4 \)
- Change in \( x \): \( 5 - 2 = 3 \)

2. From \( x = 5 \) to \( x = 8 \):
- Change in \( y \): \( -6 - (-2) = -4 \)
- Change in \( x \): \( 8 - 5 = 3 \)

The changes are consistent (the first change is -4 over a change in \( x \) of 3; the second change is also -4 over the same change in \( x \)). This indicates a potential linear relationship, as there is a constant rate of change.

To further confirm, we can calculate the slope (\( m \)):
\[
m = \frac{\Delta y}{\Delta x} = \frac{-4}{3}
\]

Since the changes in \( y \) for equal changes in \( x \) are constant, the relationship between \( x \) and \( y \) is linear.

Thus, the function that best models the data in the table is **linear**.