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)	-20	-80	-320	-1280	-5120

answer attempt 1 out of 2

this function is because
.

Explanation:

Step1: Check for common ratio

To determine if it's exponential, we check the ratio between consecutive \( f(x) \) values. Let's calculate the ratios:

• From \( x = -1 \) to \( x = 0 \): \( \frac{f(0)}{f(-1)}=\frac{-80}{-20} = 4 \)
• From \( x = 0 \) to \( x = 1 \): \( \frac{f(1)}{f(0)}=\frac{-320}{-80}=4 \)
• From \( x = 1 \) to \( x = 2 \): \( \frac{f(2)}{f(1)}=\frac{-1280}{-320} = 4 \)
• From \( x = 2 \) to \( x = 3 \): \( \frac{f(3)}{f(2)}=\frac{-5120}{-1280}=4 \)

Step2: Determine the function type

Since the ratio between consecutive \( f(x) \) values is constant (equal to 4), the function is exponential. In an exponential function, the output values have a constant common ratio, unlike linear functions which have a constant difference.

Answer:

This function is \(\boldsymbol{\text{exponential}}\) because the ratio of consecutive \( f(x) \) values is constant (equal to 4).