###### Example4.11.5

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

 $$x$$ $$y$$ $$0$$ $$-4$$ $$1$$ $$-6$$ $$2$$ $$-8$$ $$3$$ $$-10$$

We 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 decreases by $$2$$ from row to row.

 $$x$$ $$y$$ $$0$$ $$-4$$ $${}+1\rightarrow$$ $$1$$ $$-6$$ $$\leftarrow{}-2$$ $${}+1\rightarrow$$ $$2$$ $$-8$$ $$\leftarrow{}-2$$ $${}+1\rightarrow$$ $$3$$ $$-10$$ $$\leftarrow{}-2$$

Since row-to-row change is always $$1$$ for $$x$$ and is always $$-2$$ for $$y\text{,}$$ the rate of change from one row to another row is always the same: $$-2$$ units of $$y$$ for every $$1$$ unit of $$x\text{.}$$

We know that the output for $$x = 0$$ is $$y = -4\text{.}$$ And our observation about the constant rate of change tells us that if we increase the input by $$x$$ units from $$0\text{,}$$ the ouput should decrease by $$\overbrace{(-2)+(-2)+\cdots+(-2)}^{x\text{ times}}\text{,}$$ which is $$-2x\text{.}$$ So the output would be $$-4-2x\text{.}$$

So the equation is $$y=-2x-4\text{.}$$

in-context