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Sagot :
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.
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.
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