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A quadratic function is a type of polynomial function that is characterized by its highest degree of 2. It takes the general form:

• \( f(x) = ax^2 + bx + c \)

where:

• \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \)

The graph of a quadratic function is a parabola, which is a curve that can open upwards or downwards depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

The specific quadratic function from the exercise is:

• \( f(x) = -\frac{2}{5} x^2 + \frac{3}{7} x + 2 \)

This means \( a = -\frac{2}{5} \), \( b = \frac{3}{7} \), and \( c = 2 \). Because \( a < 0 \), the parabola will open downwards. Understanding the structure of a quadratic function is essential for finding its key features, such as the vertex.

The vertex of a quadratic function is a crucial point on its graph where it reaches its maximum or minimum value. This point is always on the parabola's axis of symmetry, dividing it into two mirror-image halves.
To calculate the vertex, we employ the vertex formula. For any quadratic function in the form \( ax^2 + bx + c \), the vertex \((h, k)\) has:

• \( h = -\frac{b}{2a} \)

This formula provides the x-coordinate of the vertex. Once \( h \) is found, substitute it back into the original function to determine \( k \), the y-coordinate.

• Using the given function \( f(x) = -\frac{2}{5}x^2 + \frac{3}{7}x + 2 \), we calculated \( h \) as \( \frac{15}{28} \).
• Then, plugging \( h \) back, we computed \( k \) to be \( \frac{829}{392} \).

Thus, the vertex is \( \left( \frac{15}{28}, \frac{829}{392} \right) \). Understanding the vertex formula allows you to pinpoint this critical feature of the parabola.

The parabola is the U-shaped graph that represents a quadratic function. Its orientation and width are influenced by the coefficient \( a \) in the quadratic equation:

• When \( |a| \) is large, the parabola is narrow.
• When \( |a| \) is small, the parabola is wide.

In determining the vertex of a parabola, you unveil its highest or lowest point, depending on its orientation.

For upward-opening parabolas, the vertex is at the minimum point.

For downward-opening parabolas, like in the given function \( f(x) = -\frac{2}{5} x^2 + \frac{3}{7} x + 2 \), the vertex is at the maximum point.

The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into two symmetrical halves and can be expressed with the equation \( x = h \), where \( h \) is the x-coordinate of the vertex.
Each parabola has a directrix, a horizontal line opposite from the vertex's opening and describes how 'wide' or 'narrow' the parabola is. Understanding these features of a parabola are key in graphing and analyzing quadratic functions.