2. Determine the maximum value of each quadratic expression below, and the value of \( x \) for which this maximum value exists: \( \begin{array}{ll}\text { a) }-x^{2}+4 & \text { b) }-(x-2)^{2}+5 \\ \text { c) }-(x+3)^{2}-1 & \text { d) }-2(x-1)^{2}+3\end{array} \)

To determine the maximum value of each quadratic expression and the value of \( x \) at which this maximum occurs, we can use the vertex form of a quadratic function. The general form of a quadratic function is given by: \[ f(x) = ax^2 + bx + c \] For a quadratic in the form \( f(x) = -a(x - h)^2 + k \), the maximum value occurs at \( x = h \) and the maximum value is \( k \). Let's analyze each expression step by step: ### a) \( -x^2 + 4 \) This can be rewritten in vertex form as: \[ f(x) = -1(x - 0)^2 + 4 \] - The maximum value is \( 4 \). - The value of \( x \) for which this maximum occurs is \( 0 \). ### b) \( -(x - 2)^2 + 5 \) This is already in vertex form: \[ f(x) = -1(x - 2)^2 + 5 \] - The maximum value is \( 5 \). - The value of \( x \) for which this maximum occurs is \( 2 \). ### c) \( -(x + 3)^2 - 1 \) This is also in vertex form: \[ f(x) = -1(x + 3)^2 - 1 \] - The maximum value is \( -1 \). - The value of \( x \) for which this maximum occurs is \( -3 \). ### d) \( -2(x - 1)^2 + 3 \) This is in vertex form as well: \[ f(x) = -2(x - 1)^2 + 3 \] - The maximum value is \( 3 \). - The value of \( x \) for which this maximum occurs is \( 1 \). ### Summary of Results - a) Maximum value: \( 4 \) at \( x = 0 \) - b) Maximum value: \( 5 \) at \( x = 2 \) - c) Maximum value: \( -1 \) at \( x = -3 \) - d) Maximum value: \( 3 \) at \( x = 1 \) These results provide the maximum values and corresponding \( x \) values for each quadratic expression.