Use the given information to find a formula for the exponential function N = N(t).

Question

N(3) = 27 and N(6) = 1

N(t)=

Answer 1

Jon did a lot right.

a = 729

b = 1/3

N(t) = a b^t = 729 * (1/3)^t or

729/3^t

Answer 2

If you have two points, (x₁, y₁) and (x₂, y₂), you can define the exponential function that passes through these points by substituting them in the equation y = ab^x and solving for a and b. In general, you have to solve this pair of equations: y₁ = ab^x1 and y₂ = ab^x2,

Your two equations here are:

27 = ab³ and 1 = ab^6,

^{Solve for equations for a:}

^{a = 27/}b³

a = 1/b^6,

Set those two equations in a equal to one another

27/b³  = 1/b^6,

Cross-multiplying, you can solve for b, b = 1/3

place value of b back into equation ^{a = 27/}b³

^{you will get a = 729}

^{so final answer is y = 729} b^1/3