Question

What is the exponential function for this? (-1,512)(0,128)(1,32)(2,8)(3,2)

Answer 1

The exponential function for the given points is
`\( f(x) = 2^(-x) \).`This exponential function in the form
`\( f(x) = 2^(-x) \)` aligns perfectly with the provided set of points and represents the relationship observed between the x and y values given.

Explanation:

The set of points (-1, 512), (0, 128), (1, 32), (2, 8), and (3, 2) indicates an exponential relationship in the form of
`\( f(x) = a \cdot b^x \).`By analyzing the given points, it's apparent that when x increases by 1, the y-values decrease by a factor of 4.

This indicates that the base of the exponential function is 2, as each time x increases by 1, the function value is divided by 4 (which is
`\( 2^2 \)).`The initial value of 512 at x = -1 suggests that
`\( a = 2^9 = 512 \),` hence the function is
`\( f(x) = 2^(-x) \),`producing the subsequent values of 128, 32, 8, and 2 as x increases.

This exponential function in the form
`\( f(x) = 2^(-x) \)` aligns perfectly with the provided set of points and represents the relationship observed between the x and y values given.