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Let's go through the problem step by step to build the linear model based on the table.

The table provides carbon dioxide emissions (in millions of metric tons) for various years. We are tasked with finding a linear model of the form \( y(x) = mx + b \), where:

- \( y \) represents the CO₂ emissions,
- \( x \) is the number of years past 2010 (i.e., \( x = 0 \) corresponds to 2010).

### Step (a) - Create a linear function:
To find the linear function, we first need to calculate the slope \( m \) using two data points. The slope formula is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can select any two points from the table. Let’s choose the points for 2010 and 2020:
- For 2010 (\( x = 0 \)): \( y = 336.5 \) (CO₂ emissions)
- For 2020 (\( x = 10 \)): \( y = 496.4 \) (CO₂ emissions)

\[
m = \frac{496.4 - 336.5}{10 - 0} = \frac{159.9}{10} = 15.99
\]

Now that we have the slope \( m = 15.99 \), we use the slope-intercept form \( y = mx + b \) to find \( b \), the y-intercept. Using the point for 2010 (since \( x = 0 \)):

\[
336.5 = 15.99(0) + b \quad \Rightarrow \quad b = 336.5
\]

Thus, the linear function is:

\[
y(x) = 15.99x + 336.5
\]

### Step (b) - Find the estimate for 2026:
For 2026, \( x = 2026 - 2010 = 16 \) years. Using the linear function:

\[
y(16) = 15.99(16) + 336.5 = 255.84 + 336.5 = 592.34
\]

So, the estimated CO₂ emissions in 2026 are approximately **592.34 million metric tons**.

### Step (c) - Find the slope and interpret it:
The slope \( m = 15.99 \) means that **for each year after 2010, the CO₂ emissions are expected to increase by 15.99 million metric tons**.

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Let me know if you would like further details or have any questions! Here are 5 follow-up questions you could explore:

1. What would the predicted CO₂ emissions be in 2032 using this model?
2. How accurate is the linear model compared to the data for later years like 2028 or 2030?
3. What would happen if we used other data points to create the linear function?
4. How can we check if this linear model is a good fit for the data?
5. Can this model be used to predict emissions beyond 2032? Why or why not?

**Tip:** The slope of a linear function tells you the rate of change, which is essential for understanding trends over time.

The following table gives the millions of metric tons of carbon dioxide emissions in a certain country for selected years from 2010 and projected to 2032. (a) Create a linear function that models these data, with x as the number of years past 2010 and y as the millions of metric tons of carbon dioxide emissions. (b) Find the model's estimate for the 2026 data point. (c) Find the slope of the linear model and interpret it.

Learn how to create a linear function that models carbon dioxide emissions data from 2010 to 2032, calculate the slope, and interpret the result. The task includes estimating CO₂ emissions for 2026 using the slope-intercept form. Suitable for high school students in Grades 9-12.

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