An exponential function is in the general form
#y=a(b)^x#
We know the points #(-1,8)# and #(1,2)#, so the following are true:
#8=a(b^-1)=a/b#
#2=a(b^1)=ab#
Multiply both sides of the first equation by #b# to find that
#8b=a#
Plug this into the second equation and solve for #b#:
#2=(8b)b#
#2=8b^2#
#b^2=1/4#
#b=+-1/2#
Two equations seem to be possible here. Plug both values of #b# into the either equation to find #a#. I'll use the second equation for simpler algebra.
If#b=1/2#:
#2=a(1/2)#
#a=4#
Giving us the equation: #color(green)(y=4(1/2)^x#
If#b=-1/2#:
#2=a(-1/2)#
#a=-4#
Giving us the equation: #y=-4(-1/2)^x#
However! In an exponential function, #b>0#, otherwise many issues arise when trying to graph the function.
The only valid function is
#color(green)(y=4(1/2)^x#
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