Question

Use the table of values to write the exponential function

Answer 1

Answer:

$\frac{1}{2}(\frac{1}{4}^x)$

Step-by-step explanation:

To find the function, compare the y values. Notice that each y value decreases by being divided by 4. This means the base of the exponential is 1/4.

To find the initial value, consider the point (0,0.5). When 1/4 is raised to the 0 power, the value is 1. This leaves that the initial value is 1/2 since 1/2*1 = 0.5.

The function is $\frac{1}{2}(\frac{1}{4}^x)$.

Answer 2

Answer:

The required function is $f(x)=\frac{1}{2}\left(\frac{1}{4}\right)^x$.

Step-by-step explanation:

The general exponential function is

$f(x)=ab^x$           .... (1)

where, a is the initial value and b is growth factor.

From the given table it is clear that the function passes through the points (0,0.5) and (-1,2). It means the equation of function must be satisfied by the points (0,0.5) and (-1,2).

Substitute f(x)=0.5 and x=0 in equation (1), to find the value of a.

$0.5=ab^0$

$0.5=a$

The value of a is 0.5.

Substitute a=0.5, f(x)=2 and x=-1 in equation (1), to find the value of b.

$2=(0.5)b^(-1)$

$2=\frac{0.5}{b}$

$2b=0.5$

Divide both sides by 2.

$b=\frac{0.5}{2}$

$b=0.25$

The value of b is 0.25.

Substitute a=0.5 and b=0.25 in equation (1).

$f(x)=0.5(0.25)^x$

$f(x)=\frac{1}{2}(\frac{1}{4})^x$

Therefore the required function is $f(x)=\frac{1}{2}\left(\frac{1}{4}\right)^x$.