which is the graph of $f(x)=4(\\frac{1}{2})^x$?

Answer:

The general form of an exponential - function is (y = a\cdot b^{x}), where (a) is the initial value and (b) is the base.

For the function (f(x)=4\left(\frac{1}{2}\right)^{x}), when (x = 0), (f(0)=4\left(\frac{1}{2}\right)^{0}=4\times1 = 4). So the (y) - intercept is ((0,4)).

Since (0\lt b=\frac{1}{2}\lt1), the function is a decreasing exponential function.

As (x\to+\infty), (y = 4\left(\frac{1}{2}\right)^{x}\to0), and as (x\to-\infty), (y = 4\left(\frac{1}{2}\right)^{x}\to+\infty)

We need to find the graph that has a (y) - intercept at (y = 4) and is a decreasing curve approaching the (x) - axis as (x) increases.

Explanation:

Step1: Find the y - intercept

Set (x = 0) in (y=4\left(\frac{1}{2}\right)^{x}) (y = 4\times1=4)

Step2: Determine the behavior of the function

Since (0\lt\frac{1}{2}\lt1), the function is decreasing. As (x) increases, (y) approaches (0).