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When working with quadratic functions, understanding vertex form is essential. A quadratic function written in vertex form looks like this: \( g(x) = a(x-h)^2 + k \). Here, \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards. The vertex form makes it easy to identify the vertex without needing to complete the square or use other algebraic techniques. In our example, \( g(x) = -(x-2)^2 - 4 \), we can see that \( a = -1 \), \( h = 2 \), and \( k = -4 \). Thus, the vertex of the parabola is at \( (h, k) = (2, -4) \). This means the lowest point of the parabola is at 2 on the x-axis and -4 on the y-axis.

The axis of symmetry for a parabola is a vertical line that divides the parabola into two symmetrical halves. For a quadratic function in vertex form \( g(x) = a(x-h)^2 + k \), the axis of symmetry is simply the line \( x = h \). This vertical line passes through the vertex of the parabola. It ensures that for any point on one side of the axis, there is a corresponding point on the other side at the same distance from the axis. In our function \( g(x) = -(x-2)^2 - 4 \), the axis of symmetry is \( x = 2 \). This helps us in plotting the parabola accurately and understanding its symmetry.

Quadratic functions can have either a maximum or a minimum value depending on whether the parabola opens upwards or downwards. If \( a \) (the coefficient in front of \( (x-h)^2 \) ) is positive, the parabola opens upwards and the vertex represents the minimum value of the function. If \( a \) is negative, the parabola opens downwards and the vertex represents the maximum value. In the example \( g(x) = -(x-2)^2 - 4 \), since \( a = -1 \) is negative, the parabola opens downwards. Therefore, it has a maximum value. The maximum value is simply the y-coordinate of the vertex, here it's \( k = -4 \). Thus, the highest value that \( g(x) \) can reach is -4, occurring at \( x = 2 \).