To determine which equation shows the minimum or maximum value of \( f(x) \) without changing the form of the equation, we need to identify the equations that are in vertex form, which is given by \( f(x) = a(x-h)^2 + k \). In this form, the vertex \((h, k)\) represents the maximum or minimum value of the function, depending on the sign of \( a \):
- If \( a > 0 \), the vertex represents a minimum point.
- If \( a < 0 \), the vertex represents a maximum point.
Now, let's analyze each of the options:
A. \( f(x) = -2(x+5)(x+1) \)
This is in factored form, and we would need to expand it to find the vertex.
B. \( f(x) = -2x^2 - 12x - 10 \)
This is in standard form, and we would need to convert it into vertex form to find the vertex.
C. \( f(x) = -2(x+3)^2 + 8 \)
This is in vertex form, where \( a = -2 \), \( h = -3 \), and \( k = 8 \). Since \( a < 0 \), this function has a maximum value of \( 8 \) at \( x = -3 \).
D. \( f(x) = -2(x^2 + 6x + 5) \)
This is in factored form but can be expressed in standard form by distributing the \(-2\).
From this analysis, option C is already in vertex form and clearly shows the maximum value. The others would require changes in form to identify their maximum or minimum values.
Thus, the correct answer is: C. \( f(x) = -2(x+3)^2 + 8 \)
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