Find an exponential function of the form y=ab^x whose graph passes through the points (2,1) and (3,2).
To find the exponential function of the form y = ab^x that passes through the points (2,1) and (3,2), we can substitute these points into the equation and solve for a and b.
Let's start with the point (2,1). Substitute x = 2 and y = 1 into the equation:
1 = ab^2
Next, let's substitute the other point (3,2):
2 = ab^3
Now we have a system of equations with two variables (a and b). To solve this system, we can divide the second equation by the first equation. This eliminates a and gives us an equation with only b:
(2/1) = (ab^3) / (ab^2)
2 = b
Now we know that b = 2. We can substitute this value back into one of the original equations to solve for a. Let's use the equation 1 = ab^2:
1 = ab^2
1 = a(2)^2
1 = 4a
Solve for a by dividing both sides by 4:
a = 1/4
Therefore, the exponential function that passes through the points (2,1) and (3,2) is y = (1/4)(2)^x or y = (1/4)(2^(x)).
1 = a b^2
2 = a b^3
a b^3 / a b^2 = 2
so b = 2
1 = a * 4
etc