https://file.aiquickdraw.com/mathbot/file/1728260384484sbh81fbd.jpg

The equation you are tasked with graphing is:

\[
y = x^2 - 9
\]

This is a quadratic equation, which will produce a parabola when graphed. The general form of a quadratic equation is \(y = ax^2 + bx + c\), where in your case, \(a = 1\), \(b = 0\), and \(c = -9\).

### Steps to graph the equation:

1. **Vertex**: The vertex of the parabola occurs at the point \((0, -9)\). This is because the quadratic is in the form \(y = x^2 + c\), where the vertex lies at \(x = 0\), and \(y = -9\).

2. **Plot Points**: 
    - When \(x = -3\) or \(x = 3\), \(y = 0\) (these are the x-intercepts).
    - When \(x = 2\) or \(x = -2\), \(y = -5\).
    - When \(x = 1\) or \(x = -1\), \(y = -8\).

3. **Symmetry**: Since this is a parabola, it is symmetric about the y-axis (i.e., for every point \((x, y)\), there is a corresponding point \((-x, y)\)).

4. **Graph Shape**: The parabola opens upwards because the coefficient of \(x^2\) is positive.

### Key points to plot:
- \((0, -9)\) (the vertex)
- \((3, 0)\) and \((-3, 0)\) (the x-intercepts)
- \((2, -5)\), \((-2, -5)\)
- \((1, -8)\), \((-1, -8)\)

Once you have these points, you can draw a smooth curve through them to form the parabola.

Would you like a detailed explanation on how to compute each point, or do you have any other questions?

Here are 5 questions you could consider for further exploration:
1. How do we determine the axis of symmetry for any quadratic equation?
2. What happens to the graph if we change the constant in the equation (e.g., \(y = x^2 - 5\))?
3. How can we find the vertex of a quadratic equation algebraically?
4. What is the role of the leading coefficient in determining the direction of a parabola?
5. How do we determine the x-intercepts algebraically?

**Tip**: When graphing quadratic functions, always start by finding the vertex, as it provides the center of symmetry for the parabola.

Graph the equation y = x^2 - 9 by plotting points and verify the results using a graphing utility.

Learn how to graph the quadratic equation y = x^2 - 9. This tutorial explains how to identify key points like the vertex, intercepts, and plot them to draw the parabola. Ideal for Grades 9-10 students.

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