Final answer:
To find the equation of the exponential curve that passes through the points (0,3) and (5,96), we can use the general form of an exponential equation: y = ab^x, where a is the initial value and b is the growth/decay factor. Substitute the points into the equation to get two equations. Divide the second equation by the first equation to eliminate a and solve for b. Finally, substitute the value of b back into either of the original equations to find a. Therefore, the equation of the exponential curve is y = 3(2^x).
Step-by-step explanation:
To find the equation of the exponential curve that passes through the points (0,3) and (5,96), we can use the general form of an exponential equation: y = ab^x, where a is the initial value and b is the growth/decay factor. Substitute the points into the equation to get two equations:
- 3 = ab^0 -> a = 3
- 96 = ab^5
Divide the second equation by the first equation to eliminate a and solve for b: b^5 = 96/3 = 32. Take the fifth root of both sides to get b = 2.
Finally, substitute the value of b back into either of the original equations to find a: 3 = a(2^0) = a, so a = 3.
Therefore, the equation of the exponential curve is y = 3(2^x).