To graph the piecewise function representing Isabel's shipping costs based on the weight of the books, we will first create a table from the provided data and then establish the piecewise function based on this data.
Table of Shipping Prices
| Weight Not Over (lbs.) | Price ($) | |------------------------|-----------| | 0 | 0 | | 1 | 2.69 | | 2 | 3.17 | | 3 | 3.65 | | 4 | 4.13 | | 5 | 4.61 | | 6 | 5.09 | | 7 | 5.57 | | 8 | 6.03 | | 9 | 6.49 | | 10 | 6.95 |
Piecewise Function
The piecewise function \( f(x) \) that describes the shipping price based on weight can be defined as follows:
\[ f(x) = \begin{cases} 0 & \text{if } 0 \leq x < 1 \ 2.69 & \text{if } 1 \leq x < 2 \ 3.17 & \text{if } 2 \leq x < 3 \ 3.65 & \text{if } 3 \leq x < 4 \ 4.13 & \text{if } 4 \leq x < 5 \ 4.61 & \text{if } 5 \leq x < 6 \ 5.09 & \text{if } 6 \leq x < 7 \ 5.57 & \text{if } 7 \leq x < 8 \ 6.03 & \text{if } 8 \leq x < 9 \ 6.49 & \text{if } 9 \leq x < 10 \ 6.95 & \text{if } x = 10 \end{cases} \]
Graphing the Function
To graph the piecewise function:
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Plot the Points: Start by plotting the points from the table onto the coordinate system. The x-axis will represent the weight (in pounds), and the y-axis will represent the price (in dollars).
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Draw the Segments: Connect the points with horizontal line segments (noting that they are not included for the open intervals at the end of each segment). The points at \( x = 0 \), \( x = 1 \), \( x = 2 \), ... up to \( x = 10 \) will be solid dots since they have defined prices.
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Label the Axes: Make sure to label the x-axis as "Weight (lbs.)" and the y-axis as "Price ($)".
The graph will show a step function that increases as the weight increases, reflecting the discrete pricing structure of the shipping costs.
If you would like, I can help with additional details or any specific part of the graphing process!