How do you find an exponential function given the points are (-1,8) and (1,2)?

1 Answer
Jan 9, 2016

Answer:

#y=4(1/2)^x#

Explanation:

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#