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https://file.aiquickdraw.com/mathbot/file/1729692768975f7p6r2lh.jpg

Let's analyze the questions from the images.

1. **Explain why the graph represents a function**:  
   The graph represents a function if for every input (year since 1969), there is exactly one corresponding output (carbon dioxide emission per capita). Based on the graph, it appears that for each year, there is a unique emission value, which satisfies the condition of a function.

2. **State the vertical intercept**:  
   The vertical intercept is where the graph crosses the y-axis (carbon dioxide emissions when time is 0, corresponding to the year 1969). From the graph, this intercept seems to be at around 22 tonnes of CO₂ per capita.

3. **Describe how the amount of CO₂ the average person emits is changing**:  
   The emissions start high, around 22 tonnes, and generally decrease over time. There is a dip after about 30 years, followed by a slight increase towards the end of the period shown (50 years since 1969).

4. **Evaluate C(10) and label the units**:  
   Looking at the graph at \( x = 10 \) (10 years since 1969), the corresponding emission value appears to be around 20 tonnes. Therefore, \( C(10) = 20 \) tonnes of CO₂ per capita.

5. **Using the graph, solve C(y) = 16**:  
   To solve \( C(y) = 16 \), find the point where the graph reaches 16 tonnes of CO₂. This seems to occur at about 30 years since 1969, so \( y = 30 \).

Would you like more details on any part or have further questions?

Here are 5 relative questions to delve deeper:
1. What is the ultimate definition of a function in mathematical terms?
2. Why is identifying the vertical intercept important in graph analysis?
3. How can trends in emissions be modeled using regression or other mathematical techniques?
4. What are possible real-world factors contributing to the dip and rise in CO₂ emissions after 30 years?
5. How would the interpretation change if there was more than one \( y \)-value for a single \( x \)-value on the graph?

**Tip**: Always check that the graph passes the "vertical line test" to confirm it's a function!

The graph displays how much carbon dioxide the average person in the US emits from 1969 to 2019. The questions involve interpreting the graph and applying function-related concepts.

Explore how the carbon dioxide emissions per capita in the US changed from 1969 to 2019, analyze the graph to determine key features like the vertical intercept, and solve function-based questions such as evaluating C(10) and solving C(y) = 16. Suitable for high school students.

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