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Understanding the vertex of a parabola is crucial when dealing with quadratic functions. A parabola is a U-shaped curve, and the vertex is its highest or lowest point, depending on whether it opens up or down. For the function given, \(g(x)=(x+4)^2\), the vertex form of a quadratic function is \( g(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
Here, the function is in its standard form, and it is easy to identify \(h\) and \(k\). For our function, we have \(h = -4\) and \(k = 0\). Hence, the vertex is \( (-4, 0) \).
The vertex plays an essential role in graphing because it provides a starting point and shows the parabola's optimal value point.

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in the vertex form \( g(x) = a(x-h)^2 + k \), the axis of symmetry always passes through the vertex and has the equation \( x = h \).
In our example, since the vertex is at \( (-4, 0) \), the axis of symmetry is the vertical line \( x = -4 \).
To help visualize this, we draw a dashed line through \( x = -4 \) on the graph. This line confirms the symmetry of the parabola, ensuring that every point on the left side is mirrored on the right side.

Quadratic functions are one of the most common types of polynomial functions, typically expressed in the form \( g(x) = ax^2 + bx + c \). For graphing purposes, expressing the function in vertex form, \( g(x) = a(x-h)^2 + k \), can be more useful.
In our example, \( g(x) = (x+4)^2 \), we have a quadratic function where \( a=1 \), \( h=-4 \), and \( k=0 \). This tells us that:

• The parabola opens upwards because \( a > 0 \)
• The vertex is at \( (-4, 0) \)
• The axis of symmetry is \( x = -4 \)

To graph the quadratic function, start by plotting the vertex. Since the parabola opens upwards, sketch a symmetric U-shaped curve with respect to the axis of symmetry. Quadratic functions are useful in various fields, including physics, engineering, and economics, for modeling phenomena like projectile motion and maximizing profit.