Identify the vertex and determine the minimum or maximum value of the function. \( f(x)=-2(x-4)^{2}+7 \)

The vertex of the function \( f(x) = -2(x-4)^{2} + 7 \) is found at the point \( (4, 7) \). Since the coefficient of the squared term is negative (-2), this indicates that the vertex represents a maximum point. Therefore, the maximum value of the function is \( 7 \), occurring at \( x = 4 \). To solve similar quadratic functions efficiently, remember to identify the vertex form \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. A quick check on the signs: if \( a < 0 \), you have a maximum; if \( a > 0 \), you have a minimum. This can save you time in determining the nature of the vertex! Common mistakes include overlooking the vertex form and forgetting that the sign of \( a \) determines the extremum. Always double-check your function's form and the sign of the coefficient before concluding whether you have a maximum or minimum. Happy graphing!