Answer:
[tex]f(x)=(\frac{1}{4})^{x-b}[/tex]
Step-by-step explanation:
We are given that
(0,1024),(1,256),(2,64),(3,16)and(4,4)
From given data we assume an exponential function
[tex]f(x)=a^{b+x}[/tex]
where b and a are constant
[tex]f(0)=1024[/tex]
[tex]1024=a^b[/tex]...(1)
[tex]f(1)=256[/tex]
[tex]256=a^{b+1}[/tex]...(2)
Equation (1) divided by equation (2)
Then, we get
[tex]\frac{1024}{256}=\frac{a^{b}}{a^{b+1}}[/tex]
[tex]4=a^{-1}[/tex]
[tex]a=\frac{1}{4}[/tex]
Using the the value of a in equation (1)
[tex]1024=(\frac{1}{4})^{b}[/tex]
[tex]4^{5}=4^{-b}[/tex]
[tex]-b=5[/tex]
[tex]b=-5[/tex]
Therefore, the exponential function
[tex]f(x)=(\frac{1}{4})^{x-b}[/tex]
