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Write an equation in the form $y=\ldots$ suggested by the pattern in the table.

$x$	$y$
$0$	${0}$
$1$	${1}$
$2$	${4}$
$3$	${9}$

Solution

Looking for a relationship in each row here, we see that each $y$ -value is the square of the corresponding $x$ -value. So the equation is ${y = x^{2}}\text{.}$

What if we had tried the approach we used in the previous exercise, comparing change from row to row in each column?

	$x$	$y$
	$0$	${0}$
${}+1\rightarrow$	$1$	${1}$	$\leftarrow{}+1$
${}+1\rightarrow$	$2$	${4}$	$\leftarrow{}+3$
${}+1\rightarrow$	$3$	${9}$	$\leftarrow{}+5$

Here, the rate of change is not constant from one row to the next. While the $x$ -values are increasing by $1$ from row to row, the $y$ -values increase more and more from row to row. Notice that there is a pattern there as well? Mathematicians are fascinated by relationships that produce more complicated patterns. (We’ll study more complicated patterns later.)