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When working with quadratic functions like the equation given, which is \(y=4-x^2\), creating a table of values is a very practical way of organizing your work. To start off, you should select a set of values for \(x\), usually including both positive and negative numbers, and typically centered around zero. These values allow you to see how the \(y\) values are affected by changes in \(x\).

For the exercise at hand, the values of \(x\) chosen are \(-3, -2, -1, 0, 1, 2, 3\). By substituting each value of \(x\) into the equation, you can calculate the corresponding \(y\) value. This process results in pairs of \(x, y\) coordinates that will later be used to plot the graph of the equation. The use of integer values for \(x\) makes calculating and plotting much easier, facilitating a smoother learning curve as you begin to understand the shape and direction of parabolas.

Quadratic functions are represented by the general formula \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. The graph of a quadratic function is a curve called a parabola. A key feature of parabolas is that they either open upwards or downwards, depending on the sign of the coefficient \(a\).

If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, as in our exercise where \(a = -1\), the parabola opens downwards. This distinction affects the position of the vertex, which serves as the maximum or minimum point of the graph. For the equation \(y=4-x^2\), the negative sign indicates a downward opening, and hence, the vertex will be the highest point on the graph.

The vertex of a parabola is the point where the graph changes direction. It represents the maximum or minimum value of the quadratic function. In the equation of the form \(y=ax^2+bx+c\), the vertex can be calculated using the formula \(x = -b/(2a)\).

However, the given equation \(y = 4 - x^2\) can be rewritten as \(y = -x^2 + 4\), and comparing this to the general form, we see that \(b = 0\) and \(a = -1\). This simplifies the calculation of the vertex since \(x = -0/(2*(-1)) = 0\), giving us the \(x\)-coordinate of the vertex. The \(y\)-coordinate can be found by plugging the \(x\)-coordinate of the vertex back into the original equation, which brings us to \(y = 4 - 0^2 = 4\). Therefore, the vertex of the parabola given by the equation \(y=4-x^2\) is at the point \((0, 4)\).

Plotting the graph of a quadratic function involves transferring the pairs of \(x, y\) coordinates from your table of values onto a coordinate grid. Starting by marking the vertex is often a helpful strategy, as it provides a reference point for the rest of the graph. In the given exercise, the vertex is at \((0, 4)\).

After plotting the vertex, you will plot the pairs of coordinates calculated from your table of values. Ensure that the points are plotted accurately to reflect the precise shape of the parabola. Then, draw a smooth curved line through the points, being careful to show the direction the parabola opens. In our case, the parabola opens downwards, following the negative coefficient of the \(x^2\) term. This symmetry about the vertical axis through the vertex is essential for the accurate representation of quadratic functions on a graph.