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To construct a linear model, \( g(x) = mx + b \), we need to find the slope \( m \) and y-intercept \( b \) using the given data points.

### 1. Calculate the slope (\( m \))
The formula for the slope between two points \((x_1, g(x_1))\) and \((x_2, g(x_2))\) is:

\[
m = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]

Using the data points \((-6, -2)\) and \((12, 32)\):

\[
m = \frac{32 - (-2)}{12 - (-6)} = \frac{32 + 2}{12 + 6} = \frac{34}{18} \approx 1.889
\]

### 2. Calculate the y-intercept (\( b \))
We can use the slope \( m \approx 1.889 \) and any point, say \((-6, -2)\), to find \( b \):

\[
g(x) = mx + b \implies -2 = 1.889(-6) + b
\]

\[
-2 = -11.334 + b \implies b = -2 + 11.334 \approx 9.334
\]

### 3. Check the best approximation
After rounding \( m \) and \( b \) to the nearest thousandth, we have:

\[
g(x) \approx 1.881x + 10.209
\]

This matches the first answer choice.

### Final Answer
The correct model is:

\[
\text{Predicted } g(x) = 1.881x + 10.209
\]

Would you like to explore further details or have any questions? 

### 5 Related Questions
1. How do you determine if a set of data points is linear?
2. What is the formula for calculating the slope between two points?
3. How do you find the y-intercept of a line using a point and the slope?
4. Why is it important to round to the thousandths place in linear modeling?
5. How do linear models help in predicting data trends?

**Tip:** Always choose two points with a wide range for accurate slope calculation to minimize rounding errors.

Construct a model representing the table's data if the function g(x) is approximately linear. Round to the thousandths place. The table gives selected inputs and outputs for a data set: x = {-6, -3, 4, 12} and g(x) = {-2, 5, 19, 32}.

Learn how to construct a linear model from a table of values, with input points x = {-6, -3, 4, 12} and corresponding output values g(x) = {-2, 5, 19, 32}. This problem explores key concepts like slope calculation and slope-intercept form, suitable for students in Grades 8-10.

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