Answer:
y = -2(x + 3/2)^2 + 27/2
Explanation:
Step 1: Determine the general equations of the standard and vertex forms::
The equation y = -2x^2 - 6x + 9 is in the standard form, whose general equation is given by:
y = ax^2 + bx + c
Thus, -2 is our "a" value, -6 is our "b" value, and "9" is our c value.
The general equation of the vertex form is given by:
y = a(x - h)^2 + k, where
- "a" (it's the same "a" value from the standard form) is a constant determining whether the vertex is a minimum or a maximum,
- and (h, k) are the coordinates of the vertex.
Step 2: Use the equation -b/2a to find the x-coordinate of the vertex:
The equation -b/2a comes directly from the quadratic formula and allows us to find the x-coordinate of the vertex.
Since -6 is our "b" value and -2 is our "a" value, we can find the x-coordinate of the vertex:
x-coordinate of vertex = -(-6) / 2(-2)
x-coordinate of vertex = 6 / -4
x-coordinate of vertex = -3/2
Thus, the x-coordinate of the vertex is -3/2.
Step 3: Plug in -3/2 for x to find the y-coordinate of the vertex:
Plugging in -3/2 for x will allow us to find the y-coordinate of the vertex:
y = -2(-3/2)^2 - 6(-3/2) + 9
y = -2(9/4) + 9 + 9
y = -9/2 + 9 + 9
y = 9/2 + 9
y = 27/2
Thus, the y-coordinate of the vertex is 27/2.
Step 4: Create the vertex form of y = -2x^2 - 6x + 9:
Now we can plug in -2 for a, -3/2 for h, and 27/2 for k in the vertex form, and simplifying will give us the vertex form:
y = -2(x - (-3/2))^2 + 27/2
y = -2(x + 3/2)^2 + 27/2
Thus, y = -2x^2 - 6x + 9 converted to vertex form is y = -2(x + 3/2)^2 + 27/2.