The equation \( y = 9x - x^3 \) is a cubic function. Here's a detailed analysis:

### 1. **General Shape of the Graph**
   - The function \( y = 9x - x^3 \) is a cubic polynomial. Cubic functions generally have an "S"-shaped curve.
   - The coefficient of \( x^3 \) is negative, which means the graph will fall to the left and rise to the right, creating an "S"-shape that opens downwards.

### 2. **Finding Critical Points**
   - To find the critical points (where the slope of the function is zero), take the derivative of the function with respect to \( x \):
   \[
   \frac{dy}{dx} = 9 - 3x^2
   \]
   - Set the derivative equal to zero to find the critical points:
   \[
   9 - 3x^2 = 0
   \]
   \[
   x^2 = 3
   \]
   \[
   x = \pm \sqrt{3}
   \]
   - So, the critical points occur at \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).

### 3. **Nature of Critical Points**
   - To determine whether these points are maxima or minima, evaluate the second derivative:
   \[
   \frac{d^2y}{dx^2} = -6x
   \]
   - At \( x = \sqrt{3} \), \( \frac{d^2y}{dx^2} = -6\sqrt{3} \), which is negative, indicating a local maximum.
   - At \( x = -\sqrt{3} \), \( \frac{d^2y}{dx^2} = 6\sqrt{3} \), which is positive, indicating a local minimum.

### 4. **Inflection Point**
   - The inflection point occurs where the second derivative is zero:
   \[
   -6x = 0 \implies x = 0
   \]
   - At \( x = 0 \), the function \( y = 9(0) - (0)^3 = 0 \). So, the inflection point is at \( (0, 0) \).

### 5. **Intercepts**
   - **Y-intercept**: Set \( x = 0 \) to find the y-intercept:
   \[
   y = 9(0) - (0)^3 = 0
   \]
   - The y-intercept is at \( (0, 0) \).
   - **X-intercepts**: Solve \( 9x - x^3 = 0 \):
   \[
   x(9 - x^2) = 0
   \]
   \[
   x = 0 \quad \text{or} \quad x^2 = 9 \implies x = \pm 3
   \]
   - The x-intercepts are at \( x = 0 \), \( x = 3 \), and \( x = -3 \).

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### Relative Questions:
1. What is the significance of the inflection point in a cubic function?
2. How does the coefficient of \( x^3 \) affect the shape of the cubic graph?
3. What happens to the graph if we change the coefficient of \( x \) in the equation?
4. Can you explain why the function has three x-intercepts?
5. How would you find the area under the curve of this function from \( x = -3 \) to \( x = 3 \)?

### Tip:
Understanding how to find and interpret critical points and inflection points is key to analyzing the shape and behavior of polynomial graphs.

Explore the cubic function y = 9x - x^3 in detail, including critical points, inflection points, and graph analysis. Understand how the coefficient of x^3 affects the graph shape and discover the significance of critical and inflection points. Suitable for advanced high school students and beyond.

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