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The Vertex Formula is essential in finding the peak of a quadratic function. For any quadratic function written as \( f(x) = ax^2 + bx + c \), the vertex formula is \( x = -\frac{b}{2a} \). This formula allows us to locate the x-coordinate of the vertex, which is the highest or lowest point on a parabola. It helps indicate where the function reaches its extremum, whether a maximum or minimum value.

In our example, the quadratic function is \( f(x) = -5.2x^2 - 3.8x + 5.1 \). By identifying the coefficients, we have \( a = -5.2 \), \( b = -3.8 \), and \( c = 5.1 \).

Using the vertex formula, we substitute the values into \( x = -\frac{b}{2a} \):

• \( x = -\frac{-3.8}{2(-5.2)} = \frac{3.8}{10.4} \)
• \( x \approx 0.365 \) after simplifying the arithmetic calculation

This x-coordinate of the vertex provides the critical point to find the largest or smallest y-value of the function.

Quadratic functions are known for their critical points, which are crucial to understanding their behavior. Whether the parabola represents a hill or a valley, it has either a maximum or a minimum value.

The coefficient \( a \) in the quadratic equation \( f(x) = ax^2 + bx + c \) dictates the direction of the parabola:

• If \( a > 0 \), the parabola opens upwards, featuring a minimum value.
• If \( a < 0 \), it opens downwards, presenting a maximum value.

In this case, \( a = -5.2 \), meaning the parabola opens downward, so we identify a maximum value. Once we have the x-coordinate from the vertex formula, substitute it back into the function to determine the maximum y-value:

• \( f(0.365) = -5.2 \times (0.365)^2 - 3.8 \times 0.365 + 5.1 \)
• Calculate each term: \( -5.2 \times 0.133225 \approx -0.693 \) and \( -3.8 \times 0.365 \approx -1.387 \)
• Add \(-0.693\), \(-1.387\), and \(5.1\): Approximate maximum is \(\approx 3.02\)

This calculation shows that the maximum value of the function is about \(3.02\).

A parabola is a u-shaped curve that describes the graph of a quadratic function. The orientation of a parabola—whether it opens up or down—is determined by the sign of the leading coefficient \( a \) in the function \( f(x) = ax^2 + bx + c \).

Some important characteristics of a parabola include:

• **Vertex:** The peak or trough of the parabola, where the quadratic reaches its maximum or minimum value.
• **Axis of Symmetry:** A line that passes through the vertex, dividing the parabola into two mirror-image halves.
• **Direction of Opening:** Determined by \( a \); if \( a > 0 \), it opens upwards, and if \( a < 0 \), it opens downwards.

In this function, \( f(x) = -5.2x^2 - 3.8x + 5.1 \), the leading term \( -5.2 \) tells us that the parabola opens downwards. This implies that the function has a maximum vertex point. Understanding the shape and properties of a parabola helps us predict and interpret the outcomes of applying quadratic functions to real-world situations.