91Ó°ÊÓ

In mathematics, a quadratic function is a type of polynomial function represented by the general form \( f(x) = ax^2 + bx + c \). This function is called "quadratic" because "quad" means square in Latin, and its highest degree term is a square term, \( x^2 \). Quadratic functions create a unique, symmetrical U-shaped graph known as a parabola.
A few key points about quadratic functions:

• The term \( a \) is the coefficient of \( x^2 \).
• The term \( b \) is the coefficient of \( x \).
• The term \( c \) is the constant or Y-intercept value.

Each component plays a crucial role in determining the shape and position of the parabola on a coordinate plane. Evaluating these functions involves finding values such as the vertex, axis of symmetry, and roots of the parabola. The vertex, a turning point on the graph, is particularly important as it shows the minimum or maximum point of the function depending on the parabola's orientation.

The direction in which a parabola opens is determined by the coefficient \( a \) from the quadratic function \( f(x) = ax^2 + bx + c \). This coefficient tells us whether the parabola opens upwards or downwards.

• If \( a \) is positive, the parabola opens upwards, akin to a U-shape where the vertex is the minimum point.
• If \( a \) is negative, as seen in the function \( g(x)=-x^2-6x+4 \), the parabola opens downwards, resembling an upside-down U, with the vertex acting as the maximum point.

The sign of \( a \) is crucial as it significantly influences how the parabola behaves. Regardless of its value, all parabolas exhibit symmetry around a vertical line known as the axis of symmetry, passing through the vertex.

The vertex of a parabola is a key feature that can be found without graphing by using a simple formula. For a quadratic function \( f(x) = ax^2 + bx + c \), the vertex is denoted by the coordinates \( (h, k) \), where \( h \) is the x-coordinate and \( k \) is the y-coordinate.
To determine \( h \), use the formula \( h = -\frac{b}{2a} \). Substituting the values of \( a \) and \( b \) from the quadratic equation into the formula will give you \( h \). In our example, \( h = -\frac{-6}{2(-1)} = 3 \).
Once you have \( h \), plug it back into the original quadratic equation to find \( k \). For \( g(3) \), we compute \( k = -3^2 - 6(3) + 4 = -23 \).
Now you have the vertex at \( (3, -23) \). This vertex is significant because it serves as either the peak or the lowest point, depending on the parabola's direction.