Question

Look for a pattern in each table of values to determine which kind of model best describes the data. Then write an equation for the function that models the data.

$\mathit{x}$	0	1	2	3	4
$\mathit{y}$	1	2	4	8	16

Short answer

The equation of the function that models best the given data is $\mathit{y}\mathbf{=}{\mathbf{2}}^{\mathbf{x}}$.

Step-by-step solution

Step 1. Determine the first difference, second differences or ratio of successive values.

Find the first differences, second differences or ratio of successive values as required.

First difference:

Second difference:

Ratio of successive values:

 Step 2. Write an equation for the functions that models the data. 

Since the ratios of successive $y$–values are equal; the table of values represents an exponential function of the form $y=a{b}^x$. The constant ratio is 2, so $b=2$.

Step 3. Simplify the equation for the function.

To find $a$, use an ordered pair from the table, such as $(0,1)$.

$\begin{array}{c}y=a{b}^x\\ 1=a\times 2^0[x=0,y=1]\\ 1=a\\ y=1\times 2^0\\ =2^x\end{array}$

Therefore, the equation that models the data is $y=2^x$.