Question

how do you find an exponential equation through the points (0,6) and (2, 9)?

Answer 1

an exponential equation follows the form y = ae^x + b, where and b are some constants. To use this, we plug in known values, which will give us a system of equations.
(0,6): 6 = ae^0 + b = a + b
(2,9): 9 = ae^2 + b
we can subtract the first one from the second to get:
3 = ae^2 - a = a(e^2 - 1), or a = 3/(e^2 - 1)
and we can then plug this value of a into either equation we got first to get b. I'll plug it into the first one:
6 = a + b = (3/(e^2 - 1)) + b, or b = 6 - (3/(e^2 - 1)).
Then put the values of a and b into the general form of an exponential equation
y = (3/(e^2 - 1))e^x + (6 - 3/(e^2 - 1)).
Plug the two initial points given to make sure the equation is correct