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
)
=
a
b
Multiply both sides of the first equation by
b
to find that
8
b
=
a
Plug this into the second equation and solve for
b
:
2
=
(
8
b
)
b
2
=
8
b
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:
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
y
=
4
(
1
2
)
x
