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Understanding the vertex form of a quadratic function is essential for solving problems related to maximum and minimum values. The vertex form of a quadratic function can be written as: \[ f(x) = a(x-h)^2 + k \]where \( (h, k) \) is the vertex of the parabola.

• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.

This form makes it easy to identify the vertex and the direction of the parabola. For example, in the given problem, we have: \[ f(x) = -x^2 + bx + 4 \]Since the coefficient of \( x^2 \) is negative, the parabola opens downwards, indicating a maximum value at the vertex. To convert a standard quadratic function \( ax^2 + bx + c \) into vertex form, you can complete the square, but for this exercise, understanding the vertex form helps us identify the maximum or minimum values directly.

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). There are several methods to solve them, including:

• Factoring: Expressing the quadratic as a product of linear factors, if possible.
• Completing the Square: Rewriting the quadratic in the form \( (x-h)^2 = k \).
• Quadratic Formula: Using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.

In our problem, we solved for \( b \) by converting the function to its vertex form and setting the given maximum value. This approach is also useful for finding the roots through completing the square, which aids in understanding the function's behavior better.

The properties of a parabola are crucial in understanding its shape, direction, and extreme values (maximum or minimum). Some key properties include:

• Direction: As mentioned earlier, the sign of \( a \) in \( ax^2 + bx + c \) determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).
• Vertex: The highest or lowest point on the parabola, depending on the direction.
• Axis of Symmetry: The vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. It is given by \( x = -\frac{b}{2a} \) in the standard form.
• Intercepts: Points where the parabola crosses the x-axis (roots) and y-axis. These intercepts help in graphing and understanding the parabola's behavior.

In the exercise, knowing that the parabola opens downward helped identify that the vertex represents the maximum value. This insight guided us through finding the value of \( b \).

The vertex of a parabola is a critical point providing essential information about the function's extreme values. For a given quadratic function \( ax^2 + bx + c \), the vertex can be found using: \[ x = -\frac{b}{2a} \]This formula tells us the x-coordinate of the vertex. To find the y-coordinate (the actual maximum or minimum value), substitute this x back into the original function: \[ f\left(-\frac{b}{2a}\right) \]For our problem, following these steps gave us the correct value of \( b \) that maximized the function. Understanding the vertex helps in graphing the parabola accurately and in solving optimization problems, as seen in our exercise.