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The vertex of a quadratic function is a key feature that helps us understand the graph's shape and position. For a function in the form \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

The values \( h \) and \( k \) together give us the vertex coordinates \((h, k)\). For the function given, \( f(x) = \frac{1}{2}x^2 - \frac{1}{2} \), the vertex is easily calculated because \( b = 0 \). This means \( h = 0 \).
To find \( k \), substitute \( h = 0 \) back into the function: \( k = \frac{1}{2}(0)^2 - \frac{1}{2} = -\frac{1}{2} \). Thus, the vertex is located at \((0, -\frac{1}{2})\), serving as the parabola's lowest point when \( a > 0 \).

The axis of symmetry is an imaginary line that divides a parabola into two mirror-image halves. This line runs vertically through the vertex of the quadratic function. For the quadratic equation \( ax^2 + bx + c \), the axis of symmetry can be determined using the equation:

• \( x = -\frac{b}{2a} \)

For our example, \( b = 0 \) simplifies the calculation immediately to \( x = 0 \).
Consequently, the axis of symmetry is the vertical line \( x = 0 \). This line acts like a mirror, ensuring that for every point on one side of the parabola, there is a corresponding point on the opposite side at equal distance from the axis.
It's essential for sketching a precise graph, as it helps ensure that the parabola is balanced on both sides.

A parabola is the shape of the graph of a quadratic function. It can open upwards or downwards, depending on the sign of the coefficient \( a \). In our function, \( f(x) = \frac{1}{2}x^2 - \frac{1}{2} \), \( a = \frac{1}{2} \), which is greater than zero.
This means the parabola opens upwards, creating a "U" shape. The key features of a parabola include:

• The vertex, which represents either the minimum or maximum point—here, it’s the lowest point.
• The axis of symmetry, a line that ensures the parabola is symmetrical.
• The direction of opening determined by the sign of \( a \).

The width of the parabola is also affected by \( a \)—a larger \( |a| \) makes a steeper parabola, while a smaller \( |a| \) results in a wider shape. To sketch this specific parabola, identify key points by substituting values into the function to see how it changes. The function should appear as a graceful "U" with its lowest point at the vertex.