How can we determine the initial amount and the growth factor in the exponential function g(x) = 14 * 2^x?

To identify the initial amount (a) and the growth factor (b) in the exponential function g(x) = 14 * 2^x, we can analyze the components of the function.

The general form of an exponential function is typically represented as:

g(x) = a * b^x

In this format:

• a is the initial amount, which represents the value of the function when x = 0.
• b is the growth factor, which shows how much the function multiplies for each unit increase in x.

Now, let’s match our function to this general form:

• From the function g(x) = 14 * 2^x, we can see that:
• a = 14
• b = 2

To further understand this, we can calculate the value of g(x) when x = 0:

• g(0) = 14 * 2⁰ = 14 * 1 = 14.

This confirms that the initial amount, a, is indeed 14.

For growth, since we have b = 2, it means that for every increase of 1 in x, the function will double its previous value. Thus:

• If x = 1: g(1) = 14 * 2¹ = 14 * 2 = 28
• If x = 2: g(2) = 14 * 2² = 14 * 4 = 56

This shows that the growth factor is clearly b = 2.

In summary, in the function g(x) = 14 * 2^x, the initial amount (a) is 14 and the growth factor (b) is 2.