Which type of function does this data describe? \begin{tabular}{c|ccccc}x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 144 & 72 & 36 & 18 & 9\end{tabular} A. Quadratic B. Exponential C. Linear

To determine the type of function described by the given data, let's analyze the pattern of the \( y \)-values as \( x \) changes: \[ \begin{tabular}{c|ccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 144 & 72 & 36 & 18 & 9 \end{tabular} \] **Step 1: Examine the Ratios Between Consecutive \( y \)-Values** - From \( x = -2 \) to \( x = -1 \): \( \frac{72}{144} = \frac{1}{2} \) - From \( x = -1 \) to \( x = 0 \): \( \frac{36}{72} = \frac{1}{2} \) - From \( x = 0 \) to \( x = 1 \): \( \frac{18}{36} = \frac{1}{2} \) - From \( x = 1 \) to \( x = 2 \): \( \frac{9}{18} = \frac{1}{2} \) The ratio between consecutive \( y \)-values is consistently \( \frac{1}{2} \). **Step 2: Identify the Type of Function** - **Linear Function:** Requires a constant difference between consecutive \( y \)-values. Here, the differences are halving, so it's not linear. - **Quadratic Function:** Requires a constant second difference. Calculating: - First differences: \( -72, -36, -18, -9 \) - Second differences: \( 36, 18, 9 \) (not constant) - **Exponential Function:** Characterized by a constant ratio between consecutive \( y \)-values, which matches our observations. **Conclusion:** The data follows an **exponential** pattern, specifically \( y = 36 \left(\frac{1}{2}\right)^x \). **Answer:** **B. Exponential**