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.