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

$x$	$y$
$0$	${-1}$
$1$	${2}$
$2$	${5}$
$3$	${8}$

Solution

The relationship between $x$ and $y$ in each row is not as clear here. Another popular approach for finding patterns: in each column, consider how the values change from one row to the next. From row to row, the $x$ -value increases by $1\text{.}$ Also, the $y$ -value increases by $3$ from row to row.

	$x$	$y$
	$0$	${-1}$
${}+1\rightarrow$	$1$	${2}$	$\leftarrow{}+3$
${}+1\rightarrow$	$2$	${5}$	$\leftarrow{}+3$
${}+1\rightarrow$	$3$	${8}$	$\leftarrow{}+3$

Since row-to-row change is always $1$ for $x$ and is always $3$ for $y\text{,}$ the rate of change from one row to another row is always the same: $3$ units of $y$ for every $1$ unit of $x\text{.}$

We know that the output for $x = 0$ is $y = -1\text{.}$ And our observation about the constant rate of change tells us that if we increase the input by $x$ units from $0\text{,}$ the output should increase by $\overbrace{3+3+\cdots+3}^{x\text{ times}}\text{,}$ which is $3x\text{.}$ So the output would be $-1+3x\text{.}$ So the equation is ${y = 3x-1}\text{.}$