Chapter SH, Problem 9.28EP

To determine

To find: The function represented in the given data.

Answer to Problem 9.28EP

The given data represents an exponential function.

Explanation of Solution

Given information:

The given data is shown below.

$x$	2	3	4	5	6
$y$	$\frac{9}{4}$	$\frac{27}{8}$	$\frac{81}{16}$	$\frac{243}{32}$	$\frac{729}{64}$

Calculation:

In the given table, the different value of $y$ is given for the different value of $x$ . So, the given data represents a function.

To find the relation, find the first differences of the values of $y$ .

  $\begin{array}{c}\frac{27}{8}-\frac{9}{4}=\frac{9}{8}\\ \frac{81}{16}-\frac{27}{8}=\frac{27}{16}\\ \frac{243}{32}-\frac{81}{16}=\frac{81}{32}\\ \frac{729}{64}-\frac{243}{32}=\frac{243}{64}\end{array}$

From the above differences, it is clear that the differences are not equal. So, find the second difference.

  $\begin{array}{c}\frac{27}{16}-\frac{9}{8}=\frac{9}{16}\\ \frac{81}{32}-\frac{27}{16}=\frac{27}{16}\end{array}$

From the above differences, it is clear that the differences are not equal. So, the given data does not represents a quadratic function. Determine the ratios.

  $\begin{array}{c}\frac{27}{8}÷\frac{9}{4}=\frac{3}{2}\\ \frac{81}{16}÷\frac{27}{8}=\frac{3}{2}\\ \frac{243}{32}÷\frac{81}{16}=\frac{3}{2}\\ \frac{729}{64}÷\frac{243}{32}=\frac{3}{2}\end{array}$

The ratios are equal for the successive values of $y$ . So, the given data represents an exponential function.

Therefore, the given data represents an exponential function.