The equation you provided is a quadratic function:

\[ y = x^2 - 6x + 9 \]

### Factored Form:
We can factor the quadratic expression to find the roots (or solutions) of the equation.

\[ y = (x - 3)^2 \]

This shows that the quadratic can be written as a perfect square trinomial.

### Vertex Form:
Since the equation is in the form \((x - h)^2 + k\), where \((h, k)\) is the vertex, we can identify:

- **Vertex:** \( (3, 0) \)

### Axis of Symmetry:
The axis of symmetry for the parabola is the vertical line that passes through the vertex:

\[ x = 3 \]

### Direction of the Parabola:
The parabola opens **upward** because the coefficient of \(x^2\) is positive.

### Key Points:
- **Vertex:** \((3, 0)\)
- **Y-intercept:** Set \(x = 0\) to find the y-intercept: \( y = 0^2 - 6(0) + 9 = 9 \)
- **X-intercept:** Since the vertex is on the x-axis, the x-intercept is the same as the vertex: \((3, 0)\)

Would you like more details or have any questions? Here are some related questions:

1. How do you complete the square to convert a quadratic equation to vertex form?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you graph a quadratic function given its equation?
4. Can you explain how to find the focus and directrix of a parabola?
5. What are the applications of quadratic functions in real life?

**Tip:** Always check for factoring opportunities when solving quadratic equations, as it simplifies finding the roots and understanding the graph.

Explore the quadratic function y = x^2 - 6x + 9, including factoring to find the vertex form, determining the vertex, axis of symmetry, and direction of the parabola. Ideal for students in Grades 8-10.

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