91Ó°ÊÓ

A quadratic function is a type of polynomial function that is represented by the equation \( f(x) = ax^2 + bx + c \). This equation characterizes a unique curve on a graph known as a parabola. Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \), which ensures the function is indeed quadratic.

• The term \( ax^2 \) is the quadratic term and determines the parabola's width and the direction it opens.
• The term \( bx \) is the linear term influencing the slope of the parabola.
• Constant \( c \) is the y-intercept of the graph, showing where the parabola crosses the y-axis.

Understanding the structure of a quadratic function allows us to graph the parabola and locate its features like its vertex and axis of symmetry. A quadratic function's graph is always a symmetric curve, either opening upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.

The vertex of a parabola is its highest or lowest point, depending on the direction in which the parabola opens. For a quadratic function \( f(x) = ax^2 + bx + c \), the vertex provides critical information about the graph. To find the vertex, we rely on the vertex formula:\[ x = -\frac{b}{2a} \]

• This equation gives the x-coordinate of the vertex.
• Once we have the x-coordinate, we substitute it back into the function to find the y-coordinate.

For example, in the function \( f(x) = 33x^2 - 2x + 15 \), using the formula, we find:

• The x-coordinate as \( x = -\frac{-2}{2 \times 33} = \frac{1}{33} \).
• To find the y-coordinate, substitute \( x = \frac{1}{33} \) into the function to get \( y \approx 14.9697 \).

Thus, the vertex of the parabola is the point \( \left( \frac{1}{33}, 14.9697 \right) \). This point is significant because it represents the minimum value of the function when the parabola opens upwards and the maximum when it opens downwards.

A parabola is the U-shaped graphical representation of a quadratic function. It exhibits symmetry, which is one of its defining characteristics. Whether the parabola opens upwards or downwards depends on the coefficient \( a \) in the quadratic function:

• If \( a > 0 \), the parabola opens upwards and the vertex is the minimum point.
• If \( a < 0 \), it opens downwards and the vertex is the maximum point.

The vertex acts as the tipping point of the parabola where the direction changes. Another significant aspect of a parabola is its axis of symmetry. This is an imaginary vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For the equation \( f(x) = ax^2 + bx + c \), the axis of symmetry is given by the equation \( x = \frac{-b}{2a} \).Understanding these properties helps in sketching the graph of a parabola and analyzing the behavior of quadratic functions. Whether solving optimization problems or analyzing real-world motion, parabolas play a key role across various applications.