Question

answer the questions below to determine what kind of function is depicted in the table below.
x | -1 | 0 | 1 | 2 | 3
f(x) | -6 | -24 | -96 | -384 | -1536
answer attempt 1 out of 2
this function is because
.

Explanation:

Step1: Check for common ratio

To determine if it's exponential, we find the ratio of consecutive \( f(x) \) values.
For \( x = -1 \) to \( x = 0 \): \( \frac{-24}{-6}=4 \)
For \( x = 0 \) to \( x = 1 \): \( \frac{-96}{-24}=4 \)
For \( x = 1 \) to \( x = 2 \): \( \frac{-384}{-96}=4 \)
For \( x = 2 \) to \( x = 3 \): \( \frac{-1536}{-384}=4 \)

Step2: Analyze the ratio

Since the ratio between consecutive \( f(x) \) values is constant (\( 4 \)), the function is exponential. In a linear function, the difference (not ratio) between consecutive \( f(x) \) values is constant, which is not the case here (e.g., \( -24 - (-6)= -18 \), \( -96 - (-24)= -72 \), differences are not constant).

Answer:

This function is \(\boldsymbol{\text{exponential}}\) because the ratio of consecutive \( f(x) \) values is constant (equal to \( 4 \)) (for example, \(\frac{-24}{-6} = \frac{-96}{-24}=\frac{-384}{-96}=\frac{-1536}{-384} = 4\)), indicating a constant multiplicative rate of change, characteristic of exponential functions.