Find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function.

f(x) = -2x^2 + 2x + 9
x coordinate is
y coordinate is
the equation of the line of symmetry is x =
Maximum/minimum of f(x) is
the value, f (1half) = 19 over 2 (fraction) is ?
min or max ???

To find the vertex, line of symmetry, and maximum/minimum value of the quadratic function f(x) = -2x^2 + 2x + 9:

Step 1: To find the x-coordinate of the vertex, use the formula: x = -b/2a, where a, b, and c are the coefficients of the quadratic function. In this case, a = -2 and b = 2.

x = -2/(2*(-2)) = -2/(-4) = 1/2

So, the x-coordinate of the vertex is 1/2.

Step 2: To find the y-coordinate of the vertex, substitute the x-coordinate (1/2) into the quadratic function:

f(1/2) = -2(1/2)^2 + 2(1/2) + 9
= -2(1/4) + 1 + 9
= -1/2 + 1 + 9
= 9 + 1 - 1/2
= 20 - 1/2
= 19 1/2

So, the y-coordinate of the vertex is 19 1/2.

Step 3: The equation of the line of symmetry of a quadratic function is given by x = x-coordinate of the vertex. Therefore, in this case, the equation of the line of symmetry is x = 1/2.

Step 4: To determine whether the quadratic function has a maximum or minimum value, observe the coefficient of the x^2 term. If the coefficient is negative, the function has a maximum value, and if the coefficient is positive, the function has a minimum value. In this case, the coefficient is -2, so the function has a maximum value.

Step 5: To find the maximum or minimum value of the quadratic function, substitute the x-coordinate of the vertex into the function:

f(1/2) = -2(1/2)^2 + 2(1/2) + 9
= -2(1/4) + 1 + 9
= -1/2 + 1 + 9
= 9 + 1 - 1/2
= 20 - 1/2
= 19 1/2

So, the maximum value of the quadratic function is 19 1/2.

In summary:
- The vertex of the quadratic function f(x) = -2x^2 + 2x + 9 is (1/2, 19 1/2).
- The equation of the line of symmetry is x = 1/2.
- The quadratic function has a maximum value.
- The maximum value of the quadratic function is 19 1/2.

To find the vertex, line of symmetry, and maximum/minimum value of the quadratic function f(x) = -2x^2 + 2x + 9, you can use the formula:

Vertex x-coordinate = -b / (2a)
Vertex y-coordinate = f(Vertex x-coordinate)
Line of symmetry equation: x = Vertex x-coordinate
Maximum/minimum value: y = Vertex y-coordinate

1. Find the vertex x-coordinate:
Given that the quadratic function is in the form f(x) = ax^2 + bx + c, the coefficient of x^2 is -2, and the coefficient of x is 2. Applying the formula, the vertex x-coordinate is:
Vertex x-coordinate = -b / (2a)
Vertex x-coordinate = -2 / (2 * -2)
Vertex x-coordinate = 1/2

2. Find the vertex y-coordinate:
To find the y-coordinate of the vertex, substitute the x-coordinate (-1/2) into the quadratic function:
Vertex y-coordinate = f(Vertex x-coordinate)
Vertex y-coordinate = -2(1/2)^2 + 2(1/2) + 9
Vertex y-coordinate = -1/2 + 1 + 9
Vertex y-coordinate = 19/2

Therefore, the vertex of the quadratic function f(x) = -2x^2 + 2x + 9 is (1/2 , 19/2).

3. Line of symmetry equation:
The line of symmetry is a vertical line that passes through the vertex. Its equation can be found using the x-coordinate of the vertex:
Line of symmetry equation: x = Vertex x-coordinate
Line of symmetry equation: x = 1/2

4. Maximum/minimum value:
The vertex of the quadratic function represents the maximum or minimum value of the function. In this case, since the coefficient of x^2 is negative (-2), the parabola opens downward, and the vertex represents the maximum value of the function.
Maximum value: y = Vertex y-coordinate
Maximum value: y = 19/2

5. The value f(1/2):
To find the value of f(1/2), substitute x = 1/2 into the quadratic function:
f(1/2) = -2(1/2)^2 + 2(1/2) + 9
f(1/2) = -1/2 + 1 + 9
f(1/2) = 19/2

Therefore, the value of f(1/2) is 19/2.

In summary:
- The vertex of the quadratic function is (1/2 , 19/2).
- The line of symmetry equation is x = 1/2.
- The maximum value of the function is 19/2.
- The value of f(1/2) = 19/2. Since the coefficient of x^2 is negative, the vertex represents the maximum point.