Final answer:
To find the exponential function through the points (-1, -2) and (3, -162), we set up a system of equations using the general form y = ab^x and solve for a and b. This results in the function y = 6 * (-3)^x.
Step-by-step explanation:
To find an exponential function that passes through the points (-1, -2) and (3, -162), we can use the form y = abx, where a and b are constants that we need to determine.
Substituting the first point (-1, -2) into the equation gives us -2 = a * b-1. Substituting the second point (3, -162) gives us -162 = a * b3. Now we can solve this system of equations to find the values of a and b.
To find b, we divide the second equation by the first one which gives us b4 = 81. Taking the fourth root of both sides, we find that b = 3 or b = -3. Since we are looking for a function that results in a negative y value for positive x, we choose b = -3.
Now, we can find a by substituting b = -3 into either of the original equations. Using the first equation, -2 = a * (-3)-1, we find that a = -2 * -3 = 6. Therefore, the exponential function is y = 6 * (-3)x.