Final answer:
Using the points (0,14) and (3,896) and the exponential model y = ab^x, we calculate that a = 14 and b = 4. The function y = 14(4^x) does not match any of the answer choices directly. If assuming a typo, option A might be closest if it represents the cube of 2 raised to the x power.
Step-by-step explanation:
To find the exponential function that models the data going through the points (0,14) and (3,896), we can use the general form y = ab^x. Let's use the information given in the two points to substitute into this equation and solve for a and b.
Substituting the first point (0,14) into the equation, we have y = ab^0 which simplifies to y = a. Because y = 14, we now know that a = 14.
Now, let's use the second point (3,896). We substitute into the equation to get 896 = 14b^3. To find b, we divide both sides by 14 to get b^3 = 64 and then take the cube root of both sides to find that b = 4.
Our exponential function is therefore y = 14(4^x), but this option is not listed in the choices provided. We note that 4 can be written as 2^2, which allows us to write the function as y = 14((2^2)^x) or y = 14(2^(2x)). Rewriting this in the form of y = ab^x, we see that none of the provided options are correct. However, if it's a typo and supposed to be 2^(3x), then the closest option given these points would be A. y = 14(8^x), where 8 is 2^3.