A text messaging plan charges a flat fee of $5 per month for up to 100 text messages sent plus $0.25 for each additional text message sent that month. Which of the following graphs represents the cost, y, of sending x texts in a month?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the text messaging plan and determine how the cost \( y \) changes with the number of text messages sent \( x \).

### Step-by-Step Solution:

1. **Understand the Pricing Structure**:
- The plan charges a flat fee of $5 for up to 100 text messages.
- For any text messages sent beyond 100, there is an additional charge of $0.25 for each extra message.

2. **Define the Cost Function**:
- For \( x \) (the number of text messages) less than or equal to 100:
\[ y = 5 \]
- For \( x \) greater than 100:
\[ y = 5 + 0.25(x - 100) \]
- This means that for every additional text message sent beyond 100, the cost increases by $0.25.

3. **Graph the Cost Function**:
- For \( 0 \leq x \leq 100 \), the cost \( y \) is constant at $5. This will be a horizontal line at \( y = 5 \).
- For \( x > 100 \), the cost will increase linearly. The slope of this line is \( 0.25 \), starting from the point \( (100, 5) \).

4. **Identify the Characteristics of the Graph**:
- The graph should show a horizontal line at \( y = 5 \) for \( x \) from 0 to 100.
- After \( x = 100 \), the graph should show a line that rises with a slope of \( 0.25 \).

5. **Evaluate the Given Graph Options**:
- We need to find a graph that matches the characteristics described above:
- A horizontal line at \( y = 5 \) for \( x \) from 0 to 100.
- A line that increases after \( x = 100 \).

6. **Select the Correct Graph**:
- After evaluating the options, we find that:
- Option A shows a horizontal line at \( y = 5 \) for \( x \leq 100 \) and then a line that increases after \( x = 100 \).
- Option B shows a decrease in cost after \( x = 100 \), which is not possible according to the pricing structure.

### Conclusion:
The correct graph that represents the cost \( y \) of sending \( x \) texts in a month is **Option A**.