Math Problem Statement
Which of the following graphs models the cost of parking at this parking lot, based on the piecewise function provided?
Solution
The problem asks us to determine which graph matches the function given for parking costs.
The function is:
\begin{cases} 2x & 0 \leq x \leq 4 \\ 10 & 4 < x \leq 8 \end{cases}$$ ### Step-by-step: 1. **For $$0 \leq x \leq 4$$:** The cost increases linearly at a rate of $2 per hour. For example: - At $$x = 0$$, $$y = 2(0) = 0$$. - At $$x = 2$$, $$y = 2(2) = 4$$. - At $$x = 4$$, $$y = 2(4) = 8$$. 2. **For $$4 < x \leq 8$$:** The cost is constant at $$y = 10$$. For example: - At $$x = 5, 6, 7, 8$$, $$y = 10$$. 3. **No cost is defined for $$x > 8$$:** The graph should end at $$x = 8$$. ### Evaluating Graphs: - The graph must show a line increasing from 0 to 8 for $$0 \leq x \leq 4$$, and then a flat horizontal line at $$y = 10$$ from $$x = 4$$ to $$x = 8$$. - The correct graph will reflect these behaviors. Based on this analysis, you can identify the correct graph in the options provided. If you'd like, I can confirm the correct graph visually if you need further clarification. Do you have any questions or want detailed verification? Here are related questions to expand this: 1. How do piecewise functions model real-life problems? 2. Can we explore how to graph a similar piecewise function manually? 3. What is the significance of the domain in piecewise functions? 4. How would the graph change if $$y = 3x$$ for $$0 \leq x \leq 4$$? 5. Why is the graph discontinuous after $$x > 8$$? **Tip:** Always check boundary values ($$x = 4$$) to ensure the graph transitions correctly!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Linear functions
Graph interpretation
Formulas
y = 2x for 0 ≤ x ≤ 4
y = 10 for 4 < x ≤ 8
Theorems
Basic function graphing principles
Suitable Grade Level
Grades 8-10