Exercise 9 Page 197

If the rule is linear, the rate of change any pair of points is constant. Let's use the known points from the table, $(2,\text{-}7),$ $(\text{-}3,18),$ and $(0,3)$ to investigate that proposition. Note that when we analyze rate of change, we will rewrite the points from least to greatest in terms of $x.$

By knowing the horizontal and vertical difference between the three known points we can determine the rate of change, or slope. The rate of change is constant. Therefore, this is a linear equation and it has a slope of $\text{-}5.$ We also have to determine the $y\text{-}$intercept $b,$ which occurs when the line crosses the $y\text{-}$axis at $x=0.$ From the table, we can identify this point as $(0,3).$ Now we have enough information to write the equation. Let's find the unknown values of $y.$ Finally, we can complete the table.