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A quadratic function is a type of polynomial function characterized by its highest degree of 2. This means it follows the general form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Such a function will produce a parabolic graph. Understanding the basic structure is essential:
\( a \): Coefficient of \( x^2 \), indicating the parabola's shape and orientation.
\( b \): Coefficient of \( x \), affecting the position of the vertex along the x-axis.
\( c \): Constant term, indicating the point where the parabola intersects the y-axis.
Quadratic functions are significant because they describe many real-world phenomena, from projectile motion to areas of squares and rectangles. The graph of any quadratic function is a parabola, which can open upwards (\( a > 0 \)) or downwards (\( a < 0 \)). This opening direction helps determine whether the function has a global maximum or minimum.

The vertex of a parabola is a critical point since it represents either the highest or lowest point of the graph. In a downward-opening parabola, the vertex signifies the maximum point, while in an upward-opening parabola, it's the minimum point. To find the vertex, we use the formula for the x-coordinate: \( x = -\frac{b}{2a} \).

• In our function \( g(x) = -x^2 + 4x - 5 \), \( a = -1 \) and \( b = 4 \).
• Substituting these values into the formula, we get \( x = -\frac{4}{2(-1)} = 2 \).

The x-coordinate of the vertex is 2. To find the y-coordinate, simply substitute \( x = 2 \) back into the function:
\( g(2) = 4(2) - 2^2 - 5 = -1 \).
Thus, the vertex of the parabola, \( (2, -1) \), represents the global maximum of the quadratic function \( g(x) \).

Finding the global maximum or minimum of a quadratic function depends on the parabola's direction. In our function \( g(x) = 4x - x^2 - 5 \), the parabola opens downwards because the coefficient of \( x^2 \) is negative. Hence, it has a global maximum, but not a global minimum.
When identifying the global maximum:

• Locate the vertex as it holds the maximum value (if the parabola opens downwards).
• Compute the function's value at the vertex to determine the maximum value.

For our example:
The vertex was found at \( x = 2 \) and the corresponding maximum value is \( g(2) = -1 \).
This means that the quadratic function \( g(x) = 4x - x^2 - 5 \) achieves its global maximum value of \(-1\) when \( x = 2 \). Understanding the context of a function can provide vital insights into its behavior and potential applications.