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The given quadratic function is:

\[
f(x) = - (x + 1)^2 + 9
\]

We are asked to find the zeros (roots) of this function using knowledge of quadratic shapes and transformations.

### Steps to find the zeros:
1. Set \( f(x) = 0 \) to find the points where the function crosses the x-axis:
   \[
   - (x + 1)^2 + 9 = 0
   \]
2. Isolate the squared term:
   \[
   - (x + 1)^2 = -9
   \]
   Dividing both sides by \(-1\):
   \[
   (x + 1)^2 = 9
   \]
3. Take the square root of both sides:
   \[
   x + 1 = \pm 3
   \]
4. Solve for \( x \):
   \[
   x + 1 = 3 \quad \text{or} \quad x + 1 = -3
   \]
   \[
   x = 2 \quad \text{or} \quad x = -4
   \]

### Zeros of the function:
The zeros of the function are \( x = 2 \) and \( x = -4 \).

### Explanation of the graph:
- The quadratic function is written in vertex form \( f(x) = a(x-h)^2 + k \), where the vertex is at \((h, k)\).
- Here, \( a = -1 \), meaning the parabola opens downward, and the vertex is at \( (-1, 9) \).
- Since the parabola opens downward and the zeros are \( x = 2 \) and \( x = -4 \), the graph touches the x-axis at these points.

Would you like further details on this, or do you have any questions?

Here are 5 related questions to expand on this:
1. How does the value of \( a \) in a quadratic equation affect the direction of the parabola?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How do you determine the axis of symmetry for a quadratic function?
4. How can you find the maximum or minimum value of a quadratic function?
5. How do transformations (shifts and stretches) affect the graph of a quadratic equation?

**Tip:** The discriminant of a quadratic equation can help you quickly determine whether there are real, repeated, or no real solutions.

Let f(x) = -(x + 1)^2 + 9. Use what you know about the shape of quadratic functions and their transformations to find the zeros of f(x).

Learn how to find the zeros of the quadratic function f(x) = -(x + 1)^2 + 9 by analyzing the vertex form and applying transformations. This step-by-step guide covers essential algebraic concepts and is suitable for students in Grades 9-10.

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