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A parabola is a U-shaped curve, which is the graphical representation of a quadratic function. The general form of a quadratic function is \(f(x) = ax^2 + bx + c\). When \(a\) is positive, the parabola opens upwards, and when \(a\) is negative, it opens downwards. In the function \(f(x)=x^2-4\), we are dealing with the simplest form of a quadratic equation, where \(a=1\), \(b=0\), and \(c=-4\). This results in an upward-opening parabola because \(a\) is positive.
Key features of a parabola include the vertex and the axis of symmetry. The vertex is the minimum or maximum point of the parabola. For \(f(x)=x^2-4\), the vertex is at the point (0, -4), which is also the lowest point on this graph. The axis of symmetry is a vertical line that passes through the vertex, and it indicates where the parabola is mirrored on both sides. For this function, it occurs along the line \(x = 0\).

A table of values is a helpful tool for graphing functions because it organizes the input (x-values) and corresponding output (y-values), making it easier to plot on a graph. To create a table of values for \(f(x) = x^2 - 4\), you start by selecting a range of x-values. A common choice is to pick integers around zero, such as: -3, -2, -1, 0, 1, 2, and 3.

Once you have your x-values, calculate the y-values by substituting each x into the function:\(f(x) = x^2 - 4\). For example:

• At \(x = -3\), \(f(-3)=5\).
• At \(x = 0\), \(f(0)=-4\).
• At \(x = 3\), \(f(3)=5\).

Once all the values are calculated, the table of values will help you visualize the path of the parabola by providing distinct points that you can plot.

Symmetry plays a crucial role when graphing quadratic functions, especially parabolas. For any quadratic function of the form \(f(x) = ax^2 + bx + c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\). In many cases, especially when \(b=0\) as in \(f(x)=x^2-4\), this collapses to \(x=0\), making everything symmetric about the y-axis.
This symmetry means that for every point on the left side of the parabola, there is a corresponding point on the right side an equal distance away from the axis. In our example:

• The point \((-3, 5)\) has a symmetric counterpart at \((3, 5)\).
• The point \((-1, -3)\) corresponds with \((1, -3)\).

Due to this balanced nature, once you identify several points on one side of the axis of symmetry, you can easily determine their reflections on the other side. This characteristic not only simplifies plotting but also confirms the accuracy of your graphing process.