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A quadratic function is a type of polynomial function that is distinguished by its degree, which is 2. This means the highest power of the variable is squared. Quadratic functions are commonly written in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. The graph of a quadratic function is known as a parabola, a U-shaped curve that can either open upwards or downwards.
Some key features of quadratic functions include:

• The vertex, which is the highest or lowest point on the graph.
• The axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• The direction in which the parabola opens, determined by the sign of the coefficient \( a \). If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.

The minimum value of a quadratic function is the lowest point that the function can reach, which occurs at the vertex of an upward-opening parabola. The vertex can be found using the formula for the x-coordinate: \( x = -\frac{b}{2a} \). For the given function \( y = 2x^2 - 4x + 11 \), the coefficients are \( a = 2 \) and \( b = -4 \).
Substituting these into the formula, we find:
\[ x = -\frac{-4}{2 \times 2} = \frac{4}{4} = 1 \]
Once the x-coordinate of the vertex is determined, it can be substituted back into the original quadratic equation to find the minimum y-value, which is the actual minimum output of the function. For this function at \( x = 1 \):
\[ y = 2(1)^2 - 4(1) + 11 = 9 \]
Thus, the minimum value of the function is 9, achieved when \( x = 1 \).

An upward-opening parabola is a graphical representation of a quadratic function where the coefficient \( a \) is positive. In the quadratic equation \( y = ax^2 + bx + c \), the parabola opens upwards if \( a > 0 \), forming a U-shape. This shape indicates that the vertex of the parabola is at its lowest point.
Important characteristics of an upward-opening parabola include:

• The vertex represents the minimum value of the function, as there is no value below this on the y-axis.
• The two arms of the parabola extend infinitely upwards, meaning that the function will continue to grow in magnitude as \( x \) moves further from the vertex.
• Such parabolas exhibit symmetry, with the vertex lying on the axis of symmetry, permitting the left and right sides to mirror each other.

Understanding these features is key in analyzing and solving quadratic functions, determining whether a function has a minimum or maximum value, and predicting the general behavior of the function's graph.