Answer: f(x) = 3(3)^x, where a = 3 and b = 3
Step 1: We can use the two given points to form a system of two equations and solve for the variables a and b.
- From the first point (0,3), we know that:
f(0) = ab^0 = a = 3
- From the second point (2,27), we know that:
f(2) = ab^2 = 27
Step 2: We can divide both sides of the second equation by the first equation to eliminate a and obtain:
(ab^2)/(ab^0) = 27/3
b^2 = 9
Taking the square root of both sides, we get:
b = ±3
b = 3
We need to use the positive 3 for b
Step 3: We can now substitute the value of b into one of the equations to solve for a. Let's use the first equation:
3 = a(3)^0
3 = a(1)
3 = a
Therefore, the values of a and b are a = 3 and b = 3.
So, the exponential function that passes through the points (0,3) and (2,27) is f(x) = 3(3)^x, while the value of a = 3 and the value of b = 3 also.