The graph of the function \( f(x) = -x^2 + 6x - 12 \) represents a downward-opening parabola, as indicated by the negative coefficient of the \( x^2 \) term. Let's analyze the function step-by-step:

### 1. Identifying the Parabola's Direction
The coefficient of \( x^2 \) is \(-1\), which makes the parabola open downward.

### 2. Vertex Form Conversion
To find the vertex, let's rewrite the function in vertex form by completing the square:
\[
f(x) = -x^2 + 6x - 12
\]

Rewrite \( f(x) \) as:
\[
f(x) = -(x^2 - 6x) - 12
\]

To complete the square inside the parentheses:
- Take half of \( 6 \), square it to get \( 9 \), and add \( 9 \) inside the parentheses.
\[
f(x) = -(x^2 - 6x + 9) - 12 + 9
\]
\[
f(x) = -(x - 3)^2 - 3
\]

Now, we have the vertex form of the function:
\[
f(x) = -(x - 3)^2 - 3
\]

### 3. Identifying the Vertex and Axis of Symmetry
From the vertex form \( f(x) = -(x - 3)^2 - 3 \):
- The vertex is at \( (3, -3) \).
- The axis of symmetry is \( x = 3 \).

### 4. y-Intercept
To find the y-intercept, substitute \( x = 0 \) into the function:
\[
f(0) = -(0)^2 + 6 \cdot 0 - 12 = -12
\]
So, the y-intercept is at \( (0, -12) \).

### Summary
- **Direction**: Opens downward
- **Vertex**: \( (3, -3) \)
- **Axis of Symmetry**: \( x = 3 \)
- **y-Intercept**: \( (0, -12) \)

Would you like to see the graph plotted, or do you have further questions?

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1. What are the effects of changing the leading coefficient on a parabola's direction and width?
2. How can we find the x-intercepts of this parabola?
3. What is the significance of the vertex in a quadratic function?
4. How does completing the square help in understanding parabolas?
5. Can we determine the maximum or minimum value of a quadratic function from its vertex?

**Tip:** The vertex form of a parabola, \( f(x) = a(x - h)^2 + k \), reveals both the direction and maximum/minimum point at \( (h, k) \).

Explore the graph of the quadratic function f(x) = -x^2 + 6x - 12, with step-by-step analysis. Understand key algebraic concepts like identifying the direction of parabolas, completing the square to find the vertex form, and locating intercepts. Ideal for students in Grades 9-10.

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