Question

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

$x$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$1$	$-5$	$-47$	$-161$	$-383$

answer attempt 1 out of 2
this function is because

Explanation:

Step1: Check for linear function (constant first difference)

Calculate the first differences (differences between consecutive \( f(x) \) values):

• From \( x=-1 \) to \( x = 0 \): \( -5 - 1=-6 \)
• From \( x = 0 \) to \( x = 1 \): \( -47-(-5)=-42 \)
• From \( x = 1 \) to \( x = 2 \): \( -161-(-47)=-114 \)
• From \( x = 2 \) to \( x = 3 \): \( -383-(-161)=-222 \)

First differences: \( -6, -42, -114, -222 \) (not constant, so not linear).

Step2: Check for quadratic function (constant second difference)

Calculate second differences (differences of first differences):

• \( -42-(-6)=-36 \)
• \( -114-(-42)=-72 \)
• \( -222-(-114)=-108 \)

Second differences: \( -36, -72, -108 \) (not constant, so not quadratic).

Step3: Check for cubic function (constant third difference)

Calculate third differences (differences of second differences):

• \( -72-(-36)=-36 \)
• \( -108-(-72)=-36 \)

Third differences: \( -36, -36 \) (constant). So the function is cubic (a polynomial of degree 3, since third differences are constant).

Answer:

This function is a cubic function because the third differences of the \( f(x) \) values (when calculating differences between consecutive function values) are constant.