Which exponential function passes through the points (1, 6) and (2, 12)?

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To determine which exponential function goes through the points (1, 6) and (2, 12), we start by considering the general form of an exponential function:

f(x) = a * b^x

Here, a is the initial value when x = 0, and b is the base of the exponential growth. Since we have two points, we can set up a system of equations using these points.

1. For the point (1, 6):

6 = a * b^1
         

2. For the point (2, 12):

12 = a * b^2
         

Next, we can express these equations:

  1. From the first equation, we can express a:

    2. a = 6 / b

  2. Substituting a into the second equation:

    3. 12 = (6 / b) * b^2

    This simplifies to:

    12 = 6b

    From here:

    b = 12 / 6 = 2

    Now, substituting back to find a:

    a = 6 / 2 = 3

    With a and b, the function is:

    f(x) = 3 * 2^x

    Finally, we can verify the function by plugging in the original points:

    • For (1, 6): f(1) = 3 * 2^1 = 6
    • For (2, 12): f(2) = 3 * 2^2 = 12

    Both points are verified, thus confirming that the exponential function passing through the points (1, 6) and (2, 12) is:

    f(x) = 3 * 2^x

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