To find the range of the quadratic function \( f(x) = -2(x - 4)^2 + 6 \), we first note that it is a downward-opening parabola because the coefficient of the squared term, \(-2\), is negative.
The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. In this case, we can identify:
- \( h = 4 \)
- \( k = 6 \)
This means the vertex of the parabola is at the point \((4, 6)\).
Since the parabola opens downward, the maximum value of \( f(x) \) is at the vertex, which is \(6\). Therefore, the range of the function is all values of \( y \) that are less than or equal to \(6\).
Thus, we can complete the inequality for the range of the function:
The range of the function is given by the inequality \( y \leq 6 \).
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