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The question shown asks to plot the average monthly temperature over a two-year period based on the given table, where each month is listed with corresponding temperatures. You are provided with a table showing the average monthly temperatures (in °F) for a city, and you're asked to choose the correct graph that represents this data.

Here’s the breakdown:

1. **Given Data:**
   - **Month (Jan to Dec)**: 55, 56, 57, 60, 63, 64, 65, 64, 60, 58, 55, 54 (in °F)
   - This pattern represents the temperature for **Year 1**.
   
2. **Required Task (Part a)**: 
   - Plot the data for **two years**, meaning you'll repeat the data points for another cycle, effectively doubling the period from January of the first year to December of the second year.

3. **Graph Identification**:
   - The graph must show a repeating pattern over **24 months**, starting from January (labeled as `x = 1`).
   - The plot will start low, increase through summer, peak around mid-year (June/July), and then decrease again, repeating this trend for the second year.

### Analyzing the options:

- **Option A**: This shows a cyclical pattern repeating over 24 months. It seems smooth and periodic.
- **Option B**: This graph is also periodic, but it may start lower or later in the cycle compared to others.
- **Option C**: The temperature pattern seems jagged, which may not fit the smooth transition shown by the table.
- **Option D**: This shows a pattern similar to **A**, but with a sharper rise and fall.

Based on the table's values and assuming a smooth temperature change, **Option A** appears to be the best match, as it shows a sinusoidal-like behavior that aligns with the repeating rise and fall of temperatures over two years.

Would you like more details on how I arrived at this conclusion or explanations on how to interpret these types of graphs?

### Related questions:
1. How would you create a similar graph manually using software like Excel or Desmos?
2. What is the significance of the periodic behavior in real-world temperature data?
3. How does understanding sinusoidal patterns help in interpreting temperature data?
4. Could this data fit a trigonometric model, and how would you define it?
5. What are some real-world factors that could cause deviations from this smooth temperature pattern?

### Tip:
When dealing with temperature data, always look for the **periodicity**—whether it’s monthly, yearly, or based on seasons—since it often repeats regularly due to the Earth's natural cycles.

Plot the average monthly temperature over a two-year period, letting x=1 correspond to January during the first year. Choose the correct graph from the given options.

This problem asks students to plot the average monthly temperatures for a city over a two-year period based on a table of data. The task involves recognizing periodic patterns and choosing the correct graph. The given data shows temperatures from January to December for one year, and students must replicate the pattern for the second year.

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