Answer:
f(x) = 2 -2^x
Explanation:
The graph has a horizontal asymptote at y = 2. It has a slope that doubles in each unit interval. This tells you it is an exponential function with a base of 2, and a vertical translation of 2. The curve downward, rather than upward, means the function has been reflected across a horizontal line.
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vertical shift
An exponential function of the form ...
y = a·b^x
will have a horizontal asymptote at y=0. The asymptote here is y=2, so 2 has been added to the function value.
base value
The slope of the above function (y=a·b^x) will be multiplied by a factor of 'b' when x increases by 1. Your graph has a slope (rise/run) of -1 for x between 0 and 1, a slope of -2 for x between 1 and 2, and a slope of -4 between x=2 and 3. This behavior indicates that b=-2/-1 = -4/-2 = 2.
scale factor
The function so far looks like ...
y = a·2^x +2
Using x=0, we can find the value of 'a'.
1 = a·2^0 +2
-1 = a . . . . . . . . subtract 2
Then the function is ...
f(x) = -2^x +2