Exercise 9 Page 593

Finding the Model

We want to tell whether the table of values represents a linear, exponential, or quadratic function. To do so, we will analyze how the consecutive terms are related to each other.

$x$	$\text{-}1$	$0$	$1$	$2$	$3$
$y$	$1$	$3$	$9$	$27$	$81$

Let's begin with calculating the first differences.

The first differences are not all equal. Therefore, the table of values does not represent a linear function. Let's find the second differences and compare them.

The second differences are not all equal. Therefore, the table of values does not represent a quadratic function. Let's find the ratios of the $y\text{-}$values and compare them. The ratios of successive $y\text{-}$values are equal. Therefore, the table of values can be modeled by an exponential function.

Finding the Equation

Let's recall the general form of this type of function. We will use two ordered pairs given in the table to find the values of $a$ and $b.$ For simplicity, let's use $(0,3)$ and $(1,9).$ We will start by substituting $0$ and $3$ for $x$ and $y,$ respectively. We can write a partial equation of the function represented by the table. To find the value of $b$ we will substitute $1$ for $x$ and $9$ for $y$ into our partial equation. Now we can write the equation of the function represented by the table.