To plot the graph of [tex]\( P(x) \)[/tex] according to Skylar's earnings, we need to break it down into the given intervals and define [tex]\( P(x) \)[/tex] accordingly.
The piecewise function is:
[tex]\[
P(x)=\left\{\begin{array}{ll}
15x & 0 \leq x < 2 \\
12x + 6 & 2 \leq x < 5 \\
10x + 16 & 5 \leq x \leq 8
\end{array}\right.
\][/tex]
We will plot each piece of this function on the graph.
### 1. For [tex]\(0 \leq x < 2\)[/tex]:
Here, [tex]\( P(x) = 15x \)[/tex].
- At [tex]\(x = 0\)[/tex]: [tex]\( P(0) = 15 \cdot 0 = 0 \)[/tex]
- At [tex]\(x = 2\)[/tex]: [tex]\( P(2) = 15 \cdot 2 = 30 \)[/tex]
So the segment from [tex]\( (0, 0) \)[/tex] to [tex]\( (2, 30) \)[/tex] is a straight line.
### 2. For [tex]\(2 \leq x < 5\)[/tex]:
Here, [tex]\( P(x) = 12x + 6 \)[/tex].
- At [tex]\(x = 2\)[/tex]: [tex]\( P(2) = 12 \cdot 2 + 6 = 24 + 6 = 30 \)[/tex]
- At [tex]\(x = 5\)[/tex]: [tex]\( P(5) = 12 \cdot 5 + 6 = 60 + 6 = 66 \)[/tex]
So the segment from [tex]\( (2, 30) \)[/tex] to [tex]\( (5, 66) \)[/tex] is a straight line.
### 3. For [tex]\(5 \leq x \leq 8\)[/tex]:
Here, [tex]\( P(x) = 10x + 16 \)[/tex].
- At [tex]\( x = 5 \)[/tex]: [tex]\( P(5) = 10 \cdot 5 + 16 = 50 + 16 = 66 \)[/tex]
- At [tex]\( x = 8 \)[/tex]: [tex]\( P(8) = 10 \cdot 8 + 16 = 80 + 16 = 96 \)[/tex]
So the segment from [tex]\( (5, 66) \)[/tex] to [tex]\( (8, 96) \)[/tex] is a straight line.
### Plotting the Graph
With these points calculated, we can now draw the graph [tex]\( P(x) \)[/tex]:
1. From [tex]\((0, 0)\)[/tex] to [tex]\((2, 30)\)[/tex].
2. From [tex]\((2, 30)\)[/tex] to [tex]\((5, 66)\)[/tex].
3. From [tex]\((5, 66)\)[/tex] to [tex]\((8, 96)\)[/tex].
Below is a sketch of the graph:
[tex]\[
\begin{array}{cc}
\begin{tikzpicture}[scale=1.2]
\begin{axis}[
xlabel=$x$ (hours),
ylabel={$P(x)$ (\$)},
xtick={0, 1, 2, 3, 4, 5, 6, 7, 8},
ytick={0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100},
grid=major,
legend pos=north west,
ymin=0, ymax=100,
xmin=-0.5, xmax=8.5
]
% Drawing segments
\addplot[thick, domain=0:2, samples=100] {15*x};
\addlegendentry{$P(x)=15x$}
\addplot[thick, domain=2:5, samples=100] {12*x + 6};
\addlegendentry{$P(x)=12x + 6$}
\addplot[thick, domain=5:8, samples=100] {10*x + 16};
\addlegendentry{$P(x)=10x + 16$}
\end{axis}
\end{tikzpicture}
\end{array}
\][/tex]
This graph accurately represents Skylar's pay based on the hours worked and the corresponding pay rates.
