Answer:
The required graph is shown below:
Step-by-step explanation:
Consider the provided information.
Let x represents the number of hours and S(x) represents the depth of snow.
There was no snow on the ground when it started falling at midnight at a constant rate of 1.5 inches per.
That means the depth of snow will be:
[tex]S(x)=1.5x\ \ \ 0\leq x< 4[/tex]
At 4:00 a.m., it starting falling at a constant rate of 3 inches per hour,
[tex]S(x)=3(x-4)+6\ \ \ 4\leq x<7 [/tex]
7:00 a.m. to 9:00 a.m., snow was falling at a constant rate of 2 inches per hour.
[tex]S(x)=2(x-7)+15\ \ \ 7\leq x\leq 9 [/tex]
The required piece-wise linear function is
[tex]\left\{\begin{matrix}1.5x & 0\leq x<4\\ 3(x-4)+6 & 4\leq x<7\\ 2(x-7)+15 & 7\leq x\leq 9\end{matrix}\right.[/tex]
Above function is a piece-wise linear function, so we can draw the graph by putting any 2 values of x for which the function is defined.
Substitute x=0 in [tex]S(x)=1.5x[/tex]
[tex]S(x)=1.5(0)[/tex]
[tex]S(x)=0[/tex]
Substitute x=3 in [tex]S(x)=1.5x[/tex]
[tex]S(x)=1.5(3)[/tex]
[tex]S(x)=4.5[/tex]
Now joint the points (0,0) and (3,4.5)
Remember we can't put x=4 in [tex]S(x)=1.5x[/tex] as function is not defined for x=4.
Now Substitute x=4 in [tex]S(x)=3(x-4)+6[/tex]
[tex]S(x)=3(4-4)+6[/tex]
[tex]S(x)=6[/tex]
Substitute x=5 in [tex]S(x)=3(x-4)+6[/tex]
[tex]S(x)=3(5-4)+6[/tex]
[tex]S(x)=9[/tex]
Now joint the points (4,6) and (5,9)
Now Substitute x=7 in [tex]S(x)=2(x-7)+15[/tex]
[tex]S(x)=2(7-7)+15[/tex]
[tex]S(x)=15[/tex]
Substitute x=9 in [tex]S(x)=2(x-7)+15[/tex]
[tex]S(x)=2(9-7)+15[/tex]
[tex]S(x)=19[/tex]
Now joint the points (7,15) and (9,19)
The required graph is shown below:
