a We will before we write its equation. We assume that there was no snow when the snowstorm started, making the graph start at (0,0).
Let's look at the situation sentence by sentence and add pieces to our graph as we go. We are told that the snow fell 1 inch per hour for 2 hours.
0+20+1⋅2=2 hours=2 inches of snow
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.
2+62+2⋅6=8 hours=14 inches of snow
We can add this piece to our graph.
Lastly, it snowed
1 more inch in the
final hour. That gives us the third and final piece of the function.
8+114+1⋅1=9 hours=15 inches of snow
We can add this final piece to the graph.
For each of these intervals, we can write an equation for the line in . 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 of
1. It begins at hour
0 so the is at the ,
(0,0). Let's use this to write the equation.
y=1x+0⇔y=x
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.
y=⎩⎪⎪⎨⎪⎪⎧x,Second pieceThird piece0≤x<2
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.
y=⎩⎪⎪⎨⎪⎪⎧x,y=2x−2,Third piece0≤x<22≤x<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
8th hour and lasted for
1 hour, until the storm ended. Its domain is then
8≤x≤9.
y=⎩⎪⎪⎨⎪⎪⎧x,y=2x−2,y=x+6,0≤x<22≤x<88≤x≤9