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The vertex form of a quadratic function provides an easy way to identify the vertex and other properties of the parabola. The general format is \( g(x) = a(x-h)^2 + k \). Here,

• \( a \) determines the direction and width of the parabola
• \( h \) and \( k \) represent the vertex coordinates

For example, in \( g(x) = -(x-2)^2 \), translating this into the vertex form reveals:

• \( a = -1 \), indicating the parabola opens downwards
• \( h = 2 \), which is the x-coordinate of the vertex
• \( k = 0 \), which is the y-coordinate of the vertex

So, the vertex is at \( (2, 0) \). Knowing the vertex form can make graphing and understanding the parabola much easier.

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. In vertex form, \( g(x) = a(x-h)^2 + k \), the axis of symmetry is given by the equation \( x = h \). This means it passes directly through the vertex. For \( g(x) = -(x-2)^2 \), the axis of symmetry is \( x = 2 \). This can be visualized as a dashed vertical line at \( x = 2 \), showing where the parabola will have its symmetrical behavior. Knowing where this line is helps when plotting additional points, because for any point \( (x_1, y) \) on one side, there is a corresponding point \( (x_2, y) \) with \( x_2 \) such that \( x_1 eq x_2 \) but both are equidistant from \( x = h \). This ensures the graph is accurate and well-balanced.

A parabola is a u-shaped graph that can open either upwards or downwards depending on the value of \( a \) in the quadratic function. If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards. For instance, in \( g(x) = -(x-2)^2 \), \( a = -1 \), so the parabola opens downwards. The vertex represents the highest point (if the parabola opens down) or the lowest point (if it opens up). To graph a parabola accurately, it's helpful to plot the vertex and axis of symmetry first, then use additional points to shape the curve. For example, choosing x-values and calculating corresponding y-values:

• If \( x = 0 \): \( g(0) = -(0-2)^2 = -4 \)
• If \( x = 1 \): \( g(1) = -(1-2)^2 = -1 \)

Then, use symmetry to plot points like \( (0, -4) \) alongside \( (4, -4) \) and \( (1, -1) \) along with \( (3, -1) \). Finally, connect these points with a smooth, continuous curve to illustrate the downward opening shape.