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.