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
$\mathit{y}$	0	$\mathbf{-}\mathbf{1}$	$\mathbf{-}\mathbf{4}$	$\mathbf{-}\mathbf{9}$	$\mathbf{-}\mathbf{16}$

Short answer

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

Step-by-step solution

Step 1. Determine the first and second differences.

Find the first and second differences as required.

First difference:

Second difference:

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

Since the second differences are equal, the table of values represents a quadratic function of the form $y=a{x}^2$.

Step 3. Simplify the equation for the function.

To find $a$, use an ordered pair from the table, such as $(2,-4)$.

$\begin{array}{c}y=a{x}^2\\ -4=a\times 2^2\\ 4a=-4\\ a=-1\\ y=-x^2\end{array}$

An equation that models the data is $y=-x^2$.