What does this discriminant tell you about the graph of
y = x
2 − 6x + 9?

To analyze the graph of the quadratic equation \( y = x^2 - 6x + 9 \), we can use the discriminant. The discriminant \( \Delta \) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

For the given equation \( y = x^2 - 6x + 9 \), the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = 9 \)

Substituting these values into the discriminant formula:

\[
\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0
\]

### Interpretation:
- **Discriminant \( \Delta = 0 \)**: This tells us that the quadratic equation has exactly one real solution, meaning the graph of the quadratic function touches the x-axis at exactly one point, which is called a **double root** or a **repeated root**.
- **Graph**: The graph of \( y = x^2 - 6x + 9 \) is a parabola that opens upwards and is tangent to the x-axis at this point. The vertex of the parabola is the point where it touches the x-axis.

Since the discriminant is zero, the graph will have a single point of contact with the x-axis, which is at \( x = \frac{-b}{2a} = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3 \). Substituting \( x = 3 \) back into the equation to find the y-coordinate of the vertex:

\[
y = 3^2 - 6(3) + 9 = 9 - 18 + 9 = 0
\]

Thus, the vertex of the parabola is at \( (3, 0) \), and this is the point where the graph touches the x-axis.

### Summary:
The discriminant \( \Delta = 0 \) indicates that the graph of \( y = x^2 - 6x + 9 \) is a parabola that is tangent to the x-axis at \( x = 3 \), resulting in a single real solution.

Would you like further details or have any questions?

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Here are some related questions you might consider:

1. How do you calculate the vertex of a parabola?
2. What does the discriminant tell you about the number of solutions of a quadratic equation?
3. How does the sign of the discriminant affect the shape of a parabola?
4. Can a parabola have more than two real roots?
5. What happens to the discriminant if the parabola opens downward?
6. How can you determine the axis of symmetry for a parabola?
7. What is the significance of the coefficient \( a \) in the equation \( y = ax^2 + bx + c \)?
8. How would the graph change if \( c \) were a different value in the equation?

**Tip:** Always check the discriminant first when solving quadratic equations to quickly determine the nature of the roots.

Explore the graph of y = x^2 - 6x + 9 through discriminant analysis. Learn how the discriminant indicates the number of real solutions and understand the shape of the parabola. Suitable for Grades 9-12.

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