Exercise 55 Page 224

Let's graph the first piece of the function, knowing that after $2$ hours we have $2$ inches of snow. Next, we are told that it snowed $2$ inches per hour for the following $6$ hours. This gives us the second piece of the function, which we need to add to the first piece. We can add this piece to our graph. Lastly, it snowed $1$ more inch in the $\text{final}$ $\text{hour}.$ That gives us the third and final piece of the function. We can add this final piece to the graph.

For each of these intervals, we can write an equation for the line in slope-intercept form. Let's look at this individually.

First Piece

For the first piece we know that there is $1$ inch of snow added every hour. This gives us a slope of $1.$ It begins at hour $0$ so the $y\text{-}$intercept is at the origin, $(0,0).$ Let's use this to write the equation. This describes the snowfall for the first $2$ hours. Knowing both the equation and the interval, we can write the first piece of the function.

Second Piece

The second piece represents a snowfall of $2$ inches each hour, which means that it has a slope of $2.$ We also know that it begins at the point $(2,2).$ Let's use this point to solve for $b$ in slope-intercept form. Recall that this interval began at hour $2$ and ended at hour $8.$

Third Piece

The third piece represents a snowfall of $1$ inch per hour, giving it a slope of $1.$ We can use the starting point of this piece $(8,14)$ to solve for $b$ in slope-intercept form. This rate started in the $8^{\text{th}}$ hour and lasted for $1$ hour, until the storm ended. Its domain is then $8\le x\le 9.$