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The vertex of a parabola is a crucial point that represents the peak or the lowest point of the curve, depending on its direction. In the context of the quadratic function, such as \( f(x) = ax^2 + bx + c \), the vertex provides you with a lot of valuable information about the function. To find the vertex, you can use the formula for the x-coordinate:

• \( x = -\frac{b}{2a} \)

This formula helps identify the x-position where the vertex occurs. For example, in the function \( f(x) = -3x^2 - 18x + 5 \), substituting \( a = -3 \) and \( b = -18 \) gives:

• \( x = -\frac{-18}{2(-3)} = 3 \)

Once you have the x-coordinate, you can plug it back into the function to find the y-coordinate, thus fully determining the vertex position. This single point on the graph informs whether the parabola will open upwards or downwards and is essential for understanding the behaviour of quadratic functions.

For any quadratic function, understanding whether it has a maximum or minimum value involves analyzing the coefficient of the \( x^2 \) term. When this coefficient \( a \) is negative, the parabola arches downwards, indicating a maximum value. In the function \( f(x) = -3x^2 - 18x + 5 \), the \( a \) coefficient is \( -3 \), meaning the parabola opens downwards. Consequently, there's a maximum value at the vertex of the parabola. To find the maximum value, substitute the x-coordinate of the vertex back into the function. As illustrated before, if the x-coordinate is \( 3 \), plug \( x = 3 \) into the function:

• \( f(3) = -3(3)^2 - 18(3) + 5 = -76 \)

Thus, the function's maximum value is \( -76 \). This value indicates the highest point on the parabola's curve, essential for applications like optimizing or maximizing certain scenarios in real-world contexts.

The concavity of a quadratic function helps determine the direction in which its parabola opens. If the function \( f(x) = ax^2 + bx + c \) has a leading coefficient \( a \) that is greater than zero, it opens upwards ('U' shape), indicating a minimum point. Conversely, if \( a \) is less than zero, the parabola opens downwards, indicating a maximum point.In the given quadratic function \( f(x) = -3x^2 - 18x + 5 \), the \( a \) coefficient equals \(-3\). This negative value tells us:

• The parabola opens downwards.
• There exists a maximum value.

Recognizing the concavity is essential when sketching graphs of quadratic functions, as it immediately tells you about the bounds of the function. This property is particularly useful in fields like economics, physics, and engineering, where understanding the highest or lowest points helps make informed decisions or predictions.