The equation $y = 9 - x^2$ represents a parabola in the coordinate plane.

Key features of the parabola:

1. Vertex: The vertex of the parabola is the highest point on the graph since the coefficient of $x^2$ is negative. For the equation $y = 9 - x^2$, the vertex is at $(0, 9)$.
2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. For this parabola, the axis of symmetry is $x = 0$.
3. Direction: The parabola opens downwards because the coefficient of $x^2$ is negative.
4. Intercepts:
   • Y-intercept: The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation gives $y = 9$. So, the y-intercept is $(0, 9)$.
   • X-intercepts: The x-intercepts occur when $y = 0$. Setting $y = 0$ in the equation and solving for $x$ gives: $0 = 9 - x^2 \implies x^2 = 9 \implies x = \pm 3$ So, the x-intercepts are $(-3, 0)$ and $(3, 0)$.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. How do you find the focus and directrix of the parabola $y = 9 - x^2$?
2. What is the standard form of a parabola and how does it compare to $y = 9 - x^2$?
3. How would you graph the equation $y = 9 - x^2$?
4. Can you derive the vertex form of the equation from $y = 9 - x^2$?
5. How does changing the coefficient of $x^2$ affect the shape of the parabola?

Tip: When analyzing a quadratic equation, always check the coefficient of $x^2$ to determine if the parabola opens upwards or downwards.