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What is the sign of the rate of change of y between x = 0 and x = 6?

The graph you've uploaded represents a parabola given by the equation \( y = -x^2 + 6x \), which is a downward-opening parabola. The rate of change of \( y \) as \( x \) increases from 0 to 6 refers to the slope of the tangent line to the curve, which is related to the derivative of \( y \) with respect to \( x \).

To determine the sign of the rate of change of \( y \) between \( x = 0 \) and \( x = 6 \), we need to find the derivative \( \frac{dy}{dx} \) and evaluate how it behaves in this interval.

1. **Find the derivative**:
   \[
   \frac{dy}{dx} = \frac{d}{dx}(-x^2 + 6x) = -2x + 6
   \]

2. **Evaluate the sign of the derivative**:
   The derivative \( -2x + 6 \) will tell us the slope of the tangent at any point on the curve.
   
   - At \( x = 0 \):
     \[
     \frac{dy}{dx} = -2(0) + 6 = 6 \quad (\text{positive slope, increasing})
     \]
   
   - At \( x = 6 \):
     \[
     \frac{dy}{dx} = -2(6) + 6 = -12 + 6 = -6 \quad (\text{negative slope, decreasing})
     \]

3. **Conclusion**: Between \( x = 0 \) and \( x = 6 \), the derivative starts positive and gradually decreases to become negative. This means that the function is increasing at first (positive rate of change) and then begins decreasing (negative rate of change) as \( x \) approaches 6. Thus, the rate of change is initially **positive** and then becomes **negative** before \( x = 6 \).

Would you like more detailed steps or explanations?

Here are 5 related questions to explore further:
1. How can we find the exact turning point of the parabola?
2. What is the significance of the vertex in this parabola?
3. How would the derivative behave outside the interval from \( x = 0 \) to \( x = 6 \)?
4. What does the second derivative tell us about the concavity of the curve?
5. How do we calculate the total change in \( y \) between \( x = 0 \) and \( x = 6 \)?

**Tip**: The sign of the derivative at a point tells you whether the function is increasing (positive) or decreasing (negative) at that point.

Explore the rate of change of y = -x^2 + 6x between x = 0 and x = 6. This problem covers key calculus concepts, including the derivative and how it impacts the slope of a curve. Suitable for high school students in Grades 10-12.

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