To find the range of the function \( f(x) = -(x+4)^2 + 3 \), we can analyze the function directly without the need for graphing technology.
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Identify the Type of Function: The function \( f(x) = -(x+4)^2 + 3 \) is a quadratic function. Quadratic functions are of the form \( f(x) = ax^2 + bx + c \). In this case, it opens downwards because the coefficient of the \( (x+4)^2 \) term is negative (i.e., \( a = -1 \)).
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Vertex of the Parabola: The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. Here, we see that the vertex is at \( (-4, 3) \).
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Determine the Range: Since the parabola opens downwards (as indicated by the negative coefficient), the vertex represents the maximum point of the function. The maximum value of \( f(x) \) is 3, which occurs at \( x = -4 \).
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Conclusion on the Range: As the parabola opens downwards, the function can take values from this maximum down to negative infinity. Therefore, the range of the function is:
\[ (-\infty, 3] \]
So, the final answer is: the range of the function \( f(x) = -(x+4)^2 + 3 \) is \( (-\infty, 3] \).