91Ó°ÊÓ

In mathematics, quadratic functions play a vital role in understanding different relationships and models. A quadratic function is a type of polynomial function which is characterized by its highest exponent of two, written in the form:
\[ f(x) = ax^2 + bx + c \]
where:

• \(a, b, \) and \(c\) are constants with \(a eq 0\)
• \(x\) represents the variable

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

• If \(a > 0\), the parabola opens upwards, resembling a U-shape.
• If \(a < 0\), it opens downwards, resembling an upside-down U.

Quadratic functions are often used in real-life situations like calculating projectile motion, optimizing areas, and profit maximization.

The standard form of a quadratic function is crucial for analyzing and solving quadratic equations, and it is expressed as:
\[ ax^2 + bx + c = 0 \]
where:

• \(a\) is the coefficient of the quadratic term \(x^2\)
• \(b\) is the coefficient of the linear term \(x\)
• \(c\) is the constant term

This form is incredibly useful for finding key components of the function such as the vertex, axis of symmetry, and intercepts.
Recognizing the standard form quickly allows one to apply algebraic techniques effectively, such as factoring and completing the square. For the function \( f(x) = 0.06 x^2 - 2.6 x + 3.52 \), which is already in standard form, we can directly use these features to find important aspects like the vertex using the vertex formula.

The vertex of a parabola is a critical point that indicates either the maximum or minimum value that the parabola can achieve. When a quadratic function is in standard form \( ax^2 + bx + c \), we can use the vertex formula to easily find the vertex.
The x-coordinate of the vertex is found using:
\[ x = -\frac{b}{2a} \]
This formula allows us to calculate precisely where the parabola's axis of symmetry is located. Once the x-coordinate is known, the y-coordinate can be found by plugging this value back into the original quadratic function.
For example, with \(a = 0.06\) and \(b = -2.6\), we substitute these into our vertex formula to find the x-coordinate: \[ x = -\frac{-2.6}{2 \times 0.06} = 21.67 \]
Then, we substitute \(x = 21.67\) back into the quadratic function to find the y-coordinate, thus determining that the vertex is at \((21.67, -24.678)\). This process not only gives us the vertex but also critical insight into the parabola's properties and behavior on a graph.