Question

For the following exercises, find the formula for an exponential function that passes through the two points given.

Answer 1

The formula for an exponential function that passes through the two points(-1, 3/2) and (3, 24) is determined as follows:

The formula for the exponential function is, for any real number, an exponential function is a function written in a form;

f(x) = abx

 

Here ‘a’ is the initial value, ‘b’ is any positive real number and ‘x’ is the time interval.

 

Substitute the (-1, 3/2) in the formula as follows:

3/2 = ab-1 …... (1)

 

Similarly substitute the (3,24) in the formula as follows:

24 = ab3 …... (2)

 

Use the first equation to solve for ‘a’ in terms of ‘b’:

3/2 = ab-1

3/2b = a

 

Substitute ‘a’ in the second equation to solve it for ‘b’ as follows:

          24 = (3/2b)b3

          24 = 3/2b4

24 × 2/3 = b4

          16 = b4

 

Use properties of exponents to isolate ‘b’ as follows:

b = (16)1/4

    = 2

 

Substitute the value of b in the first equation to solve it for ‘a’ as follows:

3/2(2) = a

        a = 3

 

Therefore, the exponential equation is f(x) = 3(2)x.

Therefore, the exponential equation is f(x) = 3(2)x.