Identify the exponential function of the form
f (x) = a(2 x ) + b whose
graph is shown in the figure.
a. f (x) = 3(2 x )
b.
f (x) = 2 x − 3
c.
f (x) = 2(2 x ) − 4
d. f (x) = 2 x − 2
The graph is attached a a work file.
thanks for any help
now we could go into all the theories about how to find asymptotes and how an exponential function behaves and whatnot, but here, i think it is simplest to just check the points given
on the graph, we have the points (0, -2) and (2, 1). these points will work for only one of the choices.
(0, - 2) means, when x is 0, y is -2, so let's test that
plug in x = 0 in the first, we get y = 3, so thats not our guy, so (a) is out!
plug in x = 0 in the second, we get y = -2, so this MAY be our guy, (b) is still in
plug in x = 0 in the third, we get y = -2, so (c) is still in
plug in x = 0 in the fourth, we get y = -1, so (d) is out!
Now just check (b) and (c) for the second point.
(2,1) means, when x = 2, y = 1
so for (b), plug in x = 2, we get y = 1, ah, this seems to be it!
so for (c), plug in x = 2, we get y = 4, this is not our guy!
so the answer is (c), y = 2^x - 3
NOTE: we could also realize that 2^x has a horizontal asymptote at y = 0. adding -3 shifts eveything down 3 units, so the asymptote becomes y = -3. no other graph is like that, so we could have gotten the answer that way