https://file.aiquickdraw.com/mathbot/file/17303957284148tqak5pj.jpg

Let's analyze each function in detail.

1. **Function A**: This is represented by the graph. The line in the graph has a positive slope, indicating a positive rate of change. By observing two points, \((0, 1)\) and \((4, 5)\), the slope is \(\frac{5 - 1}{4 - 0} = \frac{4}{4} = 1\). So, the equation for Function A is:
   \[
   y = x + 1
   \]
   with a rate of change (slope) of \(1\) and a starting value (y-intercept) of \(1\).

2. **Function B**: The equation is given as:
   \[
   y = -2x - 1
   \]
   Here, the rate of change (slope) is \(-2\) and the starting value (y-intercept) is \(-1\).

3. **Function C**: The description says, "y is equal to the sum of two and the product of three and x." This translates to the equation:
   \[
   y = 3x + 2
   \]
   with a rate of change (slope) of \(3\) and a starting value (y-intercept) of \(2\).

### Analyzing the Questions

1. **The function with the greatest rate of change**: Function C has the highest slope of \(3\).
2. **The function with the smallest starting value**: Function B has the smallest starting value of \(-1\).
3. **The function with the smallest rate of change**: Function A has the smallest slope of \(1\).

### Answers:

- The function with the greatest rate of change is **Function C**.
- The function with the smallest starting value is **Function B**.
- The function with the smallest rate of change is **Function A**, with a value of **1**.

Would you like more details on any of these steps, or have any questions?

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Here are five related questions to expand your understanding:

1. How does the rate of change impact the steepness of a line?
2. What does the starting value (y-intercept) represent in real-world scenarios?
3. Can you determine the slope by only looking at a graph?
4. What would happen if the slopes of two functions were the same?
5. How can you find the point of intersection between two functions?

**Tip:** The slope indicates how fast the function changes; a higher slope means a steeper line.

Compare Functions: The function with the greatest rate of change, the function with the smallest starting value, and the function with the smallest rate of change.

Learn how to compare linear functions based on rate of change and starting value. This problem covers the concepts of slope, y-intercept, and linear functions with detailed examples. Suitable for students in Grades 7-9.

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