Chapter 7.7, Problem 26PPE

To determine

Toidentify:The initial amount “ a ” and the decay factor “ b ” in given exponential function.

Answer to Problem 26PPE

The initial amount of the exponential function $y\left(x\right)=0.1\cdot {\left(0.9\right)}^x$ is $a=0.1$ and decay factor $b=0.9$ .

Explanation of Solution

Given information:

The given exponential function is $y\left(x\right)=0.1\cdot {0.9}^x$

Conceptused:

A function of the form $y\left(x\right)=a\cdot b^x$ is known as the exponential function. As $y\left(x=0\right)=a\cdot b^0=a$ , so “a” is known as initial amount of function $y\left(x\right)=a\cdot b^x$ and “ b ” is called as the growth $\left(ifb>1\right)$ of decay $\left(ifb<1\right)$ factor of function $y\left(x\right)=a\cdot b^x$ .

The given exponential function is

  $y\left(x\right)=0.1\cdot {\left(0.9\right)}^x$  ... (1)

Comparing given function (1) with the function $y\left(x\right)=a\cdot b^x$ , $a=0.1$ and $b=0.9$ .

Thus, the initial amount of the exponential function $y=0.1\cdot {\left(0.9\right)}^x$ is $a=0.1$ and the decay factor $b=0.9$ .

Conclusion:

The initial amount of the exponential function $y\left(x\right)=0.1\cdot {\left(0.9\right)}^x$ is $a=0.1$ and decay factor $b=0.9$ .