To determine which equation reveals its extreme value without needing to be altered and subsequently find its extreme value (0, 0), we should examine the given equations one by one:
A. [tex]\( y = -2x^2 - 8x \)[/tex]
This is in the standard quadratic form [tex]\( y = ax^2 + bx + c \)[/tex]. To identify the vertex directly, we need to convert it to vertex form. This equation does not reveal its extreme value without being altered.
B. [tex]\( y = 2(x + 6)(x - 2) \)[/tex]
This equation is in the factored form. To find the extreme value, we must first expand it and then convert it into a different form. Therefore, it doesn’t reveal the extreme value directly.
C. [tex]\( y = -2x^2 - 8x + 16 \)[/tex]
This equation is in the standard quadratic form and similar to equation A, it needs to be converted to the vertex form to reveal the extreme value. Hence, it doesn’t reveal its extreme value directly.
D. [tex]\( y = 2(x + 4)^2 - 2 \)[/tex]
This equation is in the vertex form [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex. In this case, [tex]\((h, k) = (-4, -2)\)[/tex]. Thus, this equation reveals the extreme value directly without any need for alterations.
Hence, the correct equation that reveals its extreme value without needing to be altered is:
Equation D
Given this form, we can see that its extreme value occurs at the vertex [tex]\((-4, -2)\)[/tex]. But the point we need to identify for extreme value is (0, 0). It's evident that the equation revealing extreme value straightforwardly in its given form is Equation D:
Equation D
