Question

Write an exponential model given two points. (7, 12) (8, 25)

Answer 1

To write an exponential model given two points, we can use the general form of an exponential function:

y = ab^x

where y is the dependent variable, x is the independent variable, b is the base of the exponential function, and a is the initial value of y when x is 0.

Using the two given points (7, 12) and (8, 25), we can form a system of equations to solve for the values of a and b:

12 = ab^7
25 = ab^8

Dividing the second equation by the first equation, we get:

25/12 = b^1

Simplifying, we get:

b = 25/12

Substituting b into the first equation, we get:

12 = a(25/12)^7

Simplifying, we get:

a = 12/(25/12)^7

Therefore, the exponential model that passes through the points (7, 12) and (8, 25) is:

y = (12/(25/12)^7)(25/12)^x

Answer 2

y = (12 / (25/12)^7) * (25/12)^x

Explanation:

To write an exponential model given two points, we can use the general form of an exponential function:

y = ab^x

where y is the dependent variable, x is the independent variable, b is the base or growth factor, and a is the initial value when x = 0.

Using the two given points, we can form a system of equations:

12 = ab^7 (1)

25 = ab^8 (2)

Dividing equation (2) by equation (1), we get:

25/12 = b^(8-7) = b

So the base of the exponential function is b = 25/12.

To find the initial value a, we can substitute b into either of the original equations. Let's use equation (1):

12 = a(25/12)^7

Simplifying, we get:

a = 12 / (25/12)^7

Therefore, the exponential model that fits the given points is:

y = (12 / (25/12)^7) * (25/12)^x