https://file.aiquickdraw.com/mathbot/file/1734707642723idd6ixcs.jpg

From the graph in the image, we are tasked with finding the **rate of change** and the **initial value** of the linear function.

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### Step 1: Identify two points on the line
From the graph, the line passes through the points:
- \( (0, 6) \) (the y-intercept)
- \( (4, 0) \) 

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### Step 2: Calculate the rate of change (slope)
The slope \( m \) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute \( (x_1, y_1) = (0, 6) \) and \( (x_2, y_2) = (4, 0) \):
\[
m = \frac{0 - 6}{4 - 0} = \frac{-6}{4} = -\frac{3}{2}
\]

Thus, the rate of change (slope) is:
\[
m = -\frac{3}{2}
\]

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### Step 3: Identify the initial value
The initial value is the \( y \)-intercept, which occurs when \( x = 0 \). From the graph, the \( y \)-intercept is \( 6 \).

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### Final Answer:
- **Rate of change:** \( -\frac{3}{2} \)
- **Initial value:** \( 6 \)

Would you like more details on how this was calculated, or any further explanations?  

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Here are 5 related questions to expand your learning:
1. How do you find the slope of a line given any two points?
2. What is the equation of this line in slope-intercept form (\(y = mx + b\))?
3. How can you determine the rate of change from a table of values instead of a graph?
4. What does a negative slope represent in a real-world context?
5. How do you find the initial value if the graph does not include the \(y\)-intercept?

**Tip:** Always double-check your points before calculating the slope to avoid sign errors!

Find the rate of change and initial value for the linear function as represented by the graph.

Learn how to find the rate of change (slope) and initial value (y-intercept) for a linear function using a graph. This problem uses points (0, 6) and (4, 0) to demonstrate how to calculate the slope as -3/2 and identify the initial value as 6. Suitable for Grade 8 students.

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