Question

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

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$	$-17$	$-51$	$-153$	$-459$	$-1377$

answer attempt 1 out of 2
this function is \\(\underline{\quad}\\) because \\(\underline{\quad}\\)

Explanation:

Step1: Check first differences

Calculate $\Delta f(x) = f(x+1)-f(x)$
$\Delta f(-2)=f(-1)-f(-2)=-51-(-17)=-34$
$\Delta f(-1)=f(0)-f(-1)=-153-(-51)=-102$
$\Delta f(0)=f(1)-f(0)=-459-(-153)=-306$
$\Delta f(1)=f(2)-f(1)=-1377-(-459)=-918$

Step2: Check ratio of first differences

Calculate $\frac{\Delta f(x+1)}{\Delta f(x)}$
$\frac{\Delta f(-1)}{\Delta f(-2)}=\frac{-102}{-34}=3$
$\frac{\Delta f(0)}{\Delta f(-1)}=\frac{-306}{-102}=3$
$\frac{\Delta f(1)}{\Delta f(0)}=\frac{-918}{-306}=3$

Step3: Match to function type

Constant ratio of differences indicates exponential function. Assume form $f(x)=ab^x+c$. Use $x=0$: $f(0)=ab^0+c=a+c=-153$. Use $x=1$: $ab+c=-459$. Subtract: $ab+c-(a+c)=a(b-1)=-306$. Use ratio $b=3$: $a(3-1)=-306 \implies 2a=-306 \implies a=-153$. Then $c=-153 - (-153)=0$. Verify $x=-2$: $-153(3^{-2})=-153\times\frac{1}{9}=-17$, which matches.

Answer:

This function is exponential because the ratio of consecutive first differences is a constant value of 3, and the values fit the exponential form $f(x)=-153(3^x)$.