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To determine the linear function from the given graph, we'll follow these steps:

### 1. Identify Two Points on the Line:
Let's pick two points where the line crosses grid intersections:

- Point 1: \((-6, -4)\)
- Point 2: \((6, 4)\)

### 2. Calculate the Slope (\(m\)):
The slope of the line is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points:
\[
m = \frac{4 - (-4)}{6 - (-6)} = \frac{4 + 4}{6 + 6} = \frac{8}{12} = \frac{2}{3}
\]

### 3. Use the Point-Slope Form to Write the Equation:
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
Using Point 1 \((-6, -4)\) and the slope \(m = \frac{2}{3}\):
\[
y - (-4) = \frac{2}{3}(x - (-6))
\]
Simplifying:
\[
y + 4 = \frac{2}{3}(x + 6)
\]
Expanding and simplifying:
\[
y + 4 = \frac{2}{3}x + 4
\]
\[
y = \frac{2}{3}x + 4 - 4
\]
\[
y = \frac{2}{3}x
\]

### Final Equation of the Line:
The linear function is:
\[
y = \frac{2}{3}x
\]

Would you like more details on this calculation, or do you have any other questions?

Here are five related questions to consider:
1. How would the equation change if the line had a negative slope?
2. How can you find the equation of a line if it's parallel to a given line?
3. What happens to the slope if the line is vertical or horizontal?
4. How do you determine the y-intercept from a graph?
5. What is the significance of the slope in the context of real-world applications?

**Tip:** When finding the slope, make sure to select points that are clearly on the grid to avoid calculation errors.

Learn how to determine the linear function from a given graph with a detailed step-by-step guide. This problem covers key concepts in algebra, including slope calculation and the point-slope form of a line. Suitable for students in Grades 9-12.

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