Question

Find the equation for an exponential function that passes through the pair of points given below. (Round all coefficients to four decimal places when necessary.) through (−1, −16) and (2, −2)

Answer 1

To find the equation for an exponential function that passes through the given points, we need to use the form of an exponential function, which is y = ab^x. We can substitute the coordinates of the points into this equation to get two equations and solve them to find the values of a and b. The equation for the exponential function that passes through the given points is y = -32(2^x).

Step-by-step explanation:

To find the equation for an exponential function that passes through the given points, we need to use the form of an exponential function, which is y = ab^x. We can substitute the coordinates of the points into this equation to get two equations: -16 = ab^(-1) and -2 = ab^2. We can then solve this system of equations to find the values of a and b.

By dividing the second equation by the first equation, we get (ab^2)/(ab^(-1)) = -2/(-16), which simplifies to b^3 = 8. Taking the cube root of both sides, we find that b = 2. Substituting this value of b into either of the original equations, we can solve for a.

Let's use the equation -16 = ab^(-1). Substituting b = 2, we get -16 = a(2^(-1)), which simplifies to -16 = a/2. Multiplying both sides by 2, we find that a = -32. Therefore, the equation for the exponential function that passes through the given points is y = -32(2^x).