https://file.aiquickdraw.com/mathbot/file/1729549589483q24an3zo.jpg

To determine which function has the greatest rate of change (slope), let's analyze each case provided.

1. **$ y = 11x - 8 $:**
   - This is a linear function in the form $ y = mx + b $, where $ m = 11 $. 
   - Therefore, the rate of change (slope) is $ 11 $.

2. **A fitness club charges a $200 membership fee plus monthly fees of $25:**
   - The total cost function can be written as $ y = 25x + 200 $, where $ 25 $ is the slope (rate of change, since the fee increases by $25 per month).
   - So, the rate of change is $ 25 $.

3. **$ y = -8x $:**
   - This is another linear equation, where the slope $ m = -8 $.
   - Therefore, the rate of change is $ -8 $.

4. **The set of points $ \{(-1, -2), (1, 2), (3, 6), (5, 10), (7, 14)\} $:**
   - To find the rate of change (slope), we can use the slope formula between any two points: 
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Let's take the first two points, $ (-1, -2) $ and $ (1, 2) $:
   \[
   m = \frac{2 - (-2)}{1 - (-1)} = \frac{4}{2} = 2
   \]
   So, the rate of change is $ 2 $.

### Conclusion:
- The function with the greatest rate of change is the **fitness club function** with a slope of $ 25 $.

Would you like more details on any specific part?

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Here are 5 related questions to expand on this concept:
1. How does the slope affect the steepness of a line on a graph?
2. Can you explain how to calculate the slope between two points in a coordinate plane?
3. What is the significance of the y-intercept in linear equations like $ y = 11x - 8 $?
4. How would the slope change if the monthly fee for the fitness club increased?
5. How do you interpret a negative slope, like in $ y = -8x $?

**Tip**: In linear equations, the larger the slope (whether positive or negative), the steeper the line will be on a graph.

Which function has the greatest rate of change? Explain. 
1. y = 11x - 8 
2. A fitness club charges a $200 membership fee plus monthly fees of $25 
3. y = -8x 
4. {(-1, -2), (1, 2), (3, 6), (5, 10), (7, 14)}

Learn how to determine the function with the greatest rate of change by analyzing different linear functions and their slopes. This includes understanding how to calculate the slope from a given equation or a set of points. Suitable for students in Grades 7-9.

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