The given equation is:

\[
y = x^2 + 9x - 3
\]

This is a quadratic equation in the form:

\[
y = ax^2 + bx + c
\]

Where:
- \( a = 1 \) (coefficient of \( x^2 \)),
- \( b = 9 \) (coefficient of \( x \)),
- \( c = -3 \) (constant term).

### Key features of the quadratic equation:

1. **Vertex**: The vertex of the parabola is the point where it changes direction. It can be found using the formula:
   \[
   x_{\text{vertex}} = -\frac{b}{2a}
   \]
   Substituting the values of \( a \) and \( b \):
   \[
   x_{\text{vertex}} = -\frac{9}{2 \times 1} = -\frac{9}{2} = -4.5
   \]
   To find the \( y \)-coordinate of the vertex, substitute \( x = -4.5 \) back into the original equation.

2. **Axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex. It is given by \( x = -\frac{b}{2a} \), which in this case is \( x = -4.5 \).

3. **Y-intercept**: The y-intercept is the point where the graph crosses the y-axis. To find this, set \( x = 0 \) in the equation:
   \[
   y = 0^2 + 9(0) - 3 = -3
   \]
   So the y-intercept is \( (0, -3) \).

4. **Direction of the parabola**: Since \( a = 1 > 0 \), the parabola opens upwards.

5. **Roots**: The solutions to \( y = 0 \) (where the graph crosses the x-axis) can be found using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Substituting \( a = 1 \), \( b = 9 \), and \( c = -3 \):
   \[
   x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-3)}}{2(1)} = \frac{-9 \pm \sqrt{81 + 12}}{2} = \frac{-9 \pm \sqrt{93}}{2}
   \]
   The roots are:
   \[
   x = \frac{-9 \pm \sqrt{93}}{2}
   \]

Would you like a more detailed explanation or help with specific points?

### Follow-up questions:
1. Would you like to solve for the exact vertex and its corresponding y-coordinate?
2. Should we simplify the roots further or calculate their approximate values?
3. Do you need help graphing this quadratic function?
4. Would you like a breakdown of how to use the quadratic formula?
5. Are you interested in how to factor this equation, if possible?

**Tip:** The discriminant \( b^2 - 4ac \) tells us the nature of the roots. If it's positive, the quadratic has two real roots.

Explore the solution to the quadratic equation y = x^2 + 9x - 3, including finding the vertex, axis of symmetry, y-intercept, and roots. Learn about the quadratic formula and vertex theorem, suitable for students in Grades 9-11.

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