Question

answer the questions below to determine what kind of function is depicted in the table below.

$x$	$-3$	$-2$	$-1$	$0$	$1$
$f(x)$	$7$	$22$	$127$	$412$	$967$

answer attempt 1 out of 2

this function is \\(\underline{\quad\quad}\\) because
\\(\underline{\quad\quad}\\).

• ratios of successive y-values are the same
• the second differences are constant
• the first differences are constant
• none of the above

Explanation:

1. First, calculate the first differences (differences between successive \( f(x) \) values):

• \( 22 - 7 = 15 \)
• \( 127 - 22 = 105 \)
• \( 412 - 127 = 285 \)
• \( 967 - 412 = 555 \)

The first differences (15, 105, 285, 555) are not constant.

1. Next, calculate the second differences (differences between successive first differences):

• \( 105 - 15 = 90 \)
• \( 285 - 105 = 180 \)
• \( 555 - 285 = 270 \)

The second differences (90, 180, 270) are not constant (they have a common difference of 90, but the second differences themselves are not constant in the sense of being the same value; they are increasing by 90 each time, so the second differences are not constant).

1. Then, calculate the ratios of successive \( f(x) \) values:

• \( \frac{22}{7}\approx3.14 \)
• \( \frac{127}{22}\approx5.77 \)
• \( \frac{412}{127}\approx3.24 \)
• \( \frac{967}{412}\approx2.35 \)

The ratios of successive \( y \)-values are not the same.

Since the first differences are not constant, the second differences are not constant, and the ratios of successive \( y \)-values are not the same, the correct option is "none of the above".

Answer:

none of the above