Find an exponential function of the form f(x)=ba^x+c with y-intercept 2, horizontal asymptote y=-2, that passe

Question

Find an exponential function of the form f(x)=ba^x+c with y-intercept 2, horizontal asymptote y=-2, that passes through the point P(1,4).
a. f(x)=-2(2^x)
b. f(x)=2(2^x) -2
c. f(x)=2(1.5^x)-2
d.f(x)=4(1.5^x)-2

Answer 1

d
You're actually given too much information for this multiple choice question.
d is the only choice which passes through (1,4), for example.
4(1.5^1) - 2 = 4*1.5 - 2 = 6-2 = 4
y-intercept at 2 means the function passes through (0,2), and d is the only one that does this as well.
(b,c and d have horizontal asymptotes of y = -2)

Answer 2

f(0) = ba^0 + c = b + c = 2
f(1) = ba^1 + c = ab + c = 4
so ab - b = 2

since the horizontal asymptote is y = -2, we know ba^x, which has y=0 as horizontal asymptote, has been translated down 2, so c = -2.

then b - 2 = 2, b = 4, and
4a - 4 = 2
4a = 6
a = 1.5

so f(x) = 4(1.5^x) - 2, answer d.