Chapter 8.6, Problem 22E

To determine

The data represent the linear, exponential or quadratic function.

Answer to Problem 22E

The function that models the given data is $f\left(x\right)={\left(\frac{1}{2}\right)}^x$

Explanation of Solution

Given information:

The given as function as shown below,

  $\left(-3,8\right),\left(-2,4\right),\left(-1,2\right),\left(0,1\right),\left(1,0.5\right)$

Formula used:

  $f\left(x\right)=a{\left(\frac{1}{2}\right)}^x$

Calculation:

Write the data in a table then find the first and second differences, when necessary, to have:

  

The data has no constant first and not second difference. Note that there is a common ratio of 0.5 that you can multiply to get the next y-value. Therefore the data must represent an exponential function with a base of 0.5 or $\frac{1}{2}$

An equation of the exponential function with a base of $\frac{1}{2}\text{or}0.5$ have:

  $f\left(x\right)=a{\left(\frac{1}{2}\right)}^x$

Use any point, say (0, 1), to find the value of a:

  $\begin{array}{l}1=a{\left(\frac{1}{2}\right)}^0\\ 1=a\left(1\right)\\ 1=a\end{array}$

Substitute the value of a into the equation to have:

  $f\left(x\right)=\left(\frac{1}{2}\right)x$

Therefore the function that models the given data is $f\left(x\right)={\left(\frac{1}{2}\right)}^x$

Conclusion:

The function that models the given data is $f\left(x\right)={\left(\frac{1}{2}\right)}^x$