Final answer:
The exponential model based on the given data points in the form y=ab^x is approximately y = 15.8489(0.7937)^x, after determining the base and the initial value a using the data points (3, 10) and (6, 5).
Step-by-step explanation:
To find the exponential model in the form y=ab^x given the data points (3, 10), (6, 5), (9, 2.5), and (12, 1.25), we observe that as the x-value increases by 3, the y-value gets divided by 2. This indicates that the base b of the exponential function is likely a fraction reflecting this decay rate. Let's use two of the points to find the model, for example, (3, 10) and (6, 5).
Formula: y = ab^x
Using point (3, 10):
10 = ab^3 (1)
Using point (6, 5):
5 = ab^6 (2)
Divide equation (2) by equation (1):
(5/10) = (ab^6)/(ab^3)
0.5 = b^3
b = ∛(0.5)
b = 0.7937 (approx)
Substitute b back into equation (1):
10 = a(0.7937)^3
a ≈ 15.8489 (approx)
The exponential model is then y = 15.8489(0.7937)^x.