Answer:
To find the values of a and b that make the power function y = ax^b pass through the points (2,5) and (6,9), we can use the following system of equations:
5 = a2^b
9 = a6^b
We need to solve for a and b in this system.
One way to do this is to divide the second equation by the first equation, which eliminates a and gives:
9/5 = (6/2)^b
Simplifying this gives:
9/5 = 3^b
Taking the logarithm of both sides (with any base) gives:
log(9/5) = log(3^b)
Using the logarithmic property that log(a^b) = b*log(a), we get:
log(9/5) = b*log(3)
Solving for b, we get:
b = log(9/5) / log(3)
Plugging this value of b into one of the original equations (e.g., the first one) gives:
5 = a*2^(log(9/5)/log(3))
Solving for a, we get:
a = 5 / 2^(log(9/5)/log(3))