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

The graph provided represents the quadratic function \( y = -x^2 + 6x \).

To determine the sign of the rate of change of \( y \) between \( x = 0 \) and \( x = 6 \), we need to analyze the behavior of the function within this interval.

### 1. Calculate the derivative of the function:

The derivative \( \frac{dy}{dx} \) represents the rate of change of \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{d}{dx} (-x^2 + 6x) = -2x + 6
\]

### 2. Analyze the sign of the derivative:
- At \( x = 0 \), \( \frac{dy}{dx} = -2(0) + 6 = 6 \). The rate of change is **positive**.
- At \( x = 6 \), \( \frac{dy}{dx} = -2(6) + 6 = -12 + 6 = -6 \). The rate of change is **negative**.

Thus, the sign of the rate of change of \( y \) transitions from positive to negative within this interval. Specifically:
- From \( x = 0 \) to \( x = 3 \) (the vertex of the parabola), the rate of change is **positive**.
- From \( x = 3 \) to \( x = 6 \), the rate of change is **negative**.

Overall, the rate of change starts positive and becomes negative as \( x \) increases from 0 to 6.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the maximum value of the function, and at which \( x \)-value does it occur?
2. How can the vertex of the parabola be found algebraically?
3. How does the derivative help identify increasing and decreasing intervals?
4. What is the significance of the sign of the second derivative in concavity analysis?
5. How would the rate of change change if the function had a different coefficient for \( x^2 \)?

**Tip:** The vertex of a parabola given by \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).

Understand how to determine the sign of the rate of change of y = -x^2 + 6x between x = 0 and x = 6 by analyzing its derivative. The solution involves key calculus concepts like derivatives and the behavior of quadratic functions.

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