Given
- an exponential curve with points (-3, 12) and (-2, 7)
- a horizontal asymptote of y=2
Find
- the equation of the curve
Solution
Since the horizontal asymptote of a plain exponential curve is y=0, your curve has been shifted up 2 units. So, we know the function is something of the form ...
... y = 2 + a·b^x
Putting in the given point values, we have
... 12 = 2 + a·b^(-3)
... 7 = 2 + a·b^(-2)
Subtracting 2 from each of these equations and finding their ratio gives
... (12-2)/(7-2) = (2+a·b^(-3) -2)/(2+a·b^(-2) -2)
... 10/5 = b^-1 . . . . . simplify
... b = 2^(-1) = 1/2
Then we can use either of the given data points to find a.
... 12 = 2 + a·(1/2)^(-3)
... 10 = 8a
... 10/8 = a = 1.25
So, the equation for the curve is ...
... y = 2 + 1.25·0.5^x
