find the vertex, line of symmetry, and minimum or maxium values of the quadratic equation

f(x)=9-x^2

F(x) = Y = -X^2 + 9.

h = Xv = -b/2a = 0/-2 = 0.

k = Yv = 0^2 + 9 = 9.

V(h,k) = (0,9).

The max. value is at the vertex and is equal to 9.

The line of symmetry = h = 0.

To find the vertex, line of symmetry, and minimum or maximum values of the quadratic equation f(x) = 9 - x^2, we can use a few key formulas and concepts.

1. Finding the vertex:
The vertex form of a quadratic equation is given by f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. To convert the given equation to vertex form, we complete the square:

f(x) = 9 - x^2
= -x^2 + 9
= -(x^2 - 9) (we factor out -1)
= -(x^2 - 2*3*x + 3^2 - 3^2) (we add and subtract the square of half of the coefficient of x)
= -(x^2 - 2*3*x + 3^2) + 9 (rearranging the terms)

From this, we can see that a = -1, h = 3, and k = 9. Therefore, the vertex of the quadratic equation is (h, k) = (3, 9).

2. Finding the line of symmetry:
The line of symmetry of a quadratic equation is given by the equation x = h, where h represents the x-coordinate of the vertex. In this case, the line of symmetry is x = 3.

3. Determining the minimum or maximum value:
In a quadratic equation in the form f(x) = a(x - h)^2 + k, if the coefficient of the squared term, a, is positive, the parabola opens upwards and the vertex represents the minimum value of the function. On the other hand, if a is negative, the parabola opens downwards and the vertex represents the maximum value of the function.

In our equation, since a = -1, the parabola opens downwards, and the vertex (3, 9) represents the maximum value of the function f(x) = 9 - x^2. Therefore, the maximum value is 9.

To summarize:
- Vertex: (3, 9)
- Line of symmetry: x = 3
- Maximum value: 9