Question

Find the equation of the exponential function represented by the table be 0 4 1 12 2 36 3 108

Answer 1

To find the equation of the exponential function represented by the given table, we need to determine the general form of an exponential function, which is given by "y = ab^x", where "a" is the initial value or y-intercept, "b" is the base or growth factor, and "x" is the exponent.

Using the table with the values:
0 4
1 12
2 36
3 108

We can determine the values of "a" and "b" by substituting the x and corresponding y values into the equation.

For x = 0, y = 4: 4 = ab^0 = a(1) = a
For x = 1, y = 12: 12 = ab^1 = ab
For x = 2, y = 36: 36 = ab^2
For x = 3, y = 108: 108 = ab^3

We can now set up a system of equations using these values:

a = 4 (from the first equation)
ab = 12 (from the second equation)
ab^2 = 36 (from the third equation)
ab^3 = 108 (from the fourth equation)

Using this system of equations, we can solve for "a" and "b".

From the second equation, we can rearrange it to get b = 12/a.

Substituting this value for b in the third equation, we get:

a(12/a)^2 = 36
12^2 = 36a
144 = 36a
a = 144/36
a = 4

So we have found that a = 4.

Now we can substitute this value for a in the second equation to solve for b:

4b = 12
b = 12/4
b = 3

So, we have determined that a = 4 and b = 3. Therefore, the equation representing the exponential function is:

y = 4 * 3^x