Exercise 24 Page 347

We want to determine whether the data show an exponential relationship. Then we will write a function that models the data. Let's do those things one at a time.

Determining the Type of the Model

If the ratios of consecutive $y\text{-}$values are equal, then the data can be modeled by an exponential function. If the difference of consecutive $y\text{-}$values is constant, then the data can be modeled by a linear function. Consider the given table.

$x$	$\text{-}20$	$\text{-}13$	$\text{-}6$	$1$	$8$	$15$
$y$	$25$	$19$	$14$	$11$	$8$	$6$

Let's calculate the difference between consecutive $y\text{-}$values. We can see that the differences are not constant, so the data cannot be modeled by a linear function. Let's determine the ratios of the consecutive $y\text{-}$values. Each ratio is around $0.75,$ so the data can be modeled by an exponential function.

Writing the Model

To find the values of $a$ and $b,$ we will use two of the ordered pairs given in the table. For simplicity, we will use $(1,11)$ and $(8,8).$ Let's start by substituting $1$ for $x$ and $11$ for $y.$ Now that we know that $a=\frac{11}{b},$ we can partially write the equation. To find the value of $b,$ we will substitute $8$ for $x$ and $8$ for $y$ in the above equation. Now that we know that $b\approx 0.96,$ we can calculate the coefficient $a=\frac{11}{b}.$ Now that we know that $a=11.5,$ we can write the full equation that models the data in the given table.