To create a piecewise function based on the bouncy house rental costs, we need to define the two scenarios presented in the problem:
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For rentals of 3 hours or less:
- The total cost \( y \) is given by the formula: \[ y = 15x + 20 \] where \( x \) is the number of hours rented (0 ≤ \( x \) ≤ 3).
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For rentals of more than 3 hours:
- The total cost \( y \) is given by the formula: \[ y = 12x \] where \( x \) is the number of hours rented ( \( x \) > 3).
Putting this together, we can write the piecewise function as:
\[ y = \begin{cases} 15x + 20 & \text{if } 0 \leq x \leq 3 \ 12x & \text{if } x > 3 \end{cases} \]
Sketching the Graph
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For 0 ≤ x ≤ 3:
- When \( x = 0 \): \( y = 15(0) + 20 = 20 \) (start at (0, 20)).
- When \( x = 1 \): \( y = 15(1) + 20 = 35 \).
- When \( x = 2 \): \( y = 15(2) + 20 = 50 \).
- When \( x = 3 \): \( y = 15(3) + 20 = 65 \) (line endpoint at (3, 65)).
- This segment of the graph is a linear equation with a slope of 15.
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For x > 3:
- At \( x = 3 \), \( y = 65 \) (this is where the line transitions).
- When \( x = 4 \): \( y = 12(4) = 48 \) (this indicates a steep drop).
- When \( x = 5 \): \( y = 12(5) = 60 \).
- When \( x = 6 \): \( y = 12(6) = 72 \).
- This segment of the graph starts at (3, 65) and is a linear equation with a slope of 12.
Summary
- From (0, 20) to (3, 65), the graph is a line with a steep incline.
- From (3, 65), the graph drops back down to (4, 48) and continues with a lower slope.
Correct Equation
The piecewise function you've graphed is:
\[ y = \begin{cases} 15x + 20 & \text{if } 0 \leq x \leq 3 \ 12x & \text{if } x > 3 \end{cases} \]
This is the function that models the total cost of renting the bouncy house based on the number of hours rented.
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