To find the maximum value of the quadratic function [tex]\( y = -2x^2 + 5 \)[/tex], we need to analyze its structure and properties.
1. Identify the quadratic function: The given function is [tex]\( y = -2x^2 + 5 \)[/tex]. Note that the coefficient of [tex]\( x^2 \)[/tex] is negative, indicating that it is a downward-facing parabola, and hence, it has a maximum value.
2. Vertex of the Parabola: For a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], the vertex of the parabola represents the maximum or minimum value. In this case, the standard form of the function given is [tex]\( y = -2x^2 + 5 \)[/tex], where [tex]\( a = -2 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 5 \)[/tex].
3. Calculate the x-coordinate of the vertex: The x-coordinate of the vertex for a parabola described by [tex]\( y = ax^2 + bx + c \)[/tex] is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
[tex]\[
x = -\frac{b}{2a} = -\frac{0}{2(-2)} = 0
\][/tex]
4. Find the y-coordinate of the vertex: To find the maximum value of the function, substitute the x-coordinate of the vertex back into the original quadratic equation [tex]\( y = -2x^2 + 5 \)[/tex].
[tex]\[
y = -2(0)^2 + 5 = 5
\][/tex]
5. Conclusion: The y-coordinate of the vertex is the highest point of the parabola, which represents the maximum value of the function.
Therefore, the maximum value of the function [tex]\( y = -2x^2 + 5 \)[/tex] is [tex]\( \boxed{5} \)[/tex].
