Question

Find the equation for an exponential function that passes through the points (2, 48) and (4, 768).

Answer 1

f(x) = a(b^x)

f(2) = ab² = 48

f(4) = ab⁴ = 768

b² = 768/48 = 16

b = 4, a = 3

f(x) = 3(4^x)

Answer 2

To find the equation for an exponential function that passes through the points (2, 48) and (4, 768), we can use the general form of an exponential function and solve for the values of a and b. The equation for the exponential function that passes through the given points is y = (1/12)(24)^x.

Step-by-step explanation:

To find the equation for an exponential function that passes through the points (2, 48) and (4, 768), we can use the general form of an exponential function, which is y = ab^x. We need to determine the values of a and b.

Substituting the first point (2, 48), we have 48 = ab^2. Substituting the second point (4, 768), we have 768 = ab^4.

1. Using the first equation, divide both sides by a to isolate b^2: b^2 = 48/a.
2. Substituting this expression for b^2 into the second equation, we have 768 = a(48/a)^2. Simplifying, we get 768 = 48^2/a.
3. To solve for a, multiply both sides by a: 768a = 48^2.
4. Divide both sides by 48^2 to solve for a: a = 768/(48^2) = 1/12.
5. Substituting the value of a back into the first equation, we have 48 = (1/12)b^2. Solving for b^2, we get b^2 = 12*48 = 576.
6. Taking the square root of both sides, we get b = ±24.

Therefore, the equation for the exponential function that passes through the given points is y = (1/12)(24)^x.