To determine which exponential function passes through the points (1, 6) and (2, 12), we start by using the general form of an exponential function, which is given by:
f(x) = a * b^x
Here, a is a constant, and b is the base of the exponential function.
We know that:
- f(1) = 6, which gives us the equation: a * b^1 = 6
- f(2) = 12, which gives us the equation: a * b^2 = 12
Now we have a system of equations:
- 1. a * b = 6
- 2. a * b^2 = 12
From the first equation, we can express a in terms of b:
a = 6/b
Now, substitute this expression for a into the second equation:
(6/b) * b^2 = 12
Simplifying this, we get:
6b = 12
From this, we find:
b = 2
Now, substituting b back into the first equation to find a:
a * 2 = 6
Thus:
a = 3
So, the exponential function that goes through the points (1, 6) and (2, 12) is:
f(x) = 3 * 2^x
To verify, we can plug in the values:
- For x = 1: f(1) = 3 * 2^1 = 6
- For x = 2: f(2) = 3 * 2^2 = 12
Both points check out, confirming that the correct exponential function is indeed f(x) = 3 * 2^x.