Write a piecewise function that models the cellular phone billing plan described below. Let x represent the number of minutes used and​ C(x) represent the cost for those x minutes. Then graph the function.
​$25.00
 
per month buys 350
 
minutes.  Additional time costs $ 0.25
 
per minute.
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Part 1
Fill in the missing values or expressions for the piecewise function below.
Upper C left parenthesis x right parenthesis equals
	
enter your response here
if 0 less than or equals x less than or equals nothing
enter your response here
 
▼
 
x less than 350
x greater than 350 x greater than 25.00 x less than 25.00
​(Simplify your answer. Do not include the​ $ symbol in your​ answer.)

To model the given cellular phone billing plan using a piecewise function, we need to account for two scenarios:

1. If the number of minutes used $ x $ is less than or equal to 350 minutes, the cost is a flat $25.00.
2. If the number of minutes used $ x $ exceeds 350 minutes, the cost is $25.00 for the first 350 minutes, plus $0.25 for each additional minute.

Let's define the piecewise function for $ C(x) $, where $ C(x) $ represents the cost for $ x $ minutes.

### Piecewise Function:
\[
C(x) =
\begin{cases} 
25 & \text{if } 0 \leq x \leq 350 \\
25 + 0.25(x - 350) & \text{if } x > 350
\end{cases}
\]

### Explanation:
- For $ 0 \leq x \leq 350 $, the cost is a flat rate of $25.00.
- For $ x > 350 $, the cost starts at $25.00 and increases by $0.25 for each minute beyond 350.

### Graph:
- The graph would show a horizontal line at $ C(x) = 25 $ for $ 0 \leq x \leq 350 $.
- After $ x = 350 $, the graph becomes a straight line with a slope of 0.25, indicating the additional cost per minute.

Would you like me to graph this function for you?

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Here are 5 related questions to explore this topic further:

1. How does the slope of the function change after 350 minutes?
2. What would happen to the function if the per-minute charge after 350 minutes increased?
3. How would the graph look if the flat rate was for 400 minutes instead of 350?
4. What happens if the user uses exactly 350 minutes? How much will they pay?
5. How can you express the function if the per-minute charge was variable rather than fixed?

**Tip:** Piecewise functions are useful to represent real-world scenarios where different conditions apply over different intervals.

Write a piecewise function that models the cellular phone billing plan described below. Let x represent the number of minutes used and​ C(x) represent the cost for those x minutes. Then graph the function. $25.00 per month buys 350 minutes. Additional time costs $0.25 per minute.

Learn how to model a cellular phone billing plan using a piecewise function. The plan includes a flat rate of $25 for up to 350 minutes and an additional charge of $0.25 per minute after 350 minutes. Explore the mathematical concepts and formulas behind this real-world cost modeling problem, suitable for Grades 9-12.

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