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Quadratic functions are a class of polynomial functions that take the general form of \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The graph of a quadratic function is called a parabola. It can either open upwards when \(a > 0\) or downwards when \(a < 0\).

For the given function \(f(x) = x^2 + 2\), it is a simplified quadratic function with no \(x\)-term, meaning \(b = 0\), and with \(a = 1\) and \(c = 2\). This basic form reveals directly that the parabola opens upwards because \(a\) is positive and that it will be shifted vertically 2 units from the origin, which is understood in the process of graphing. Graphing a quadratic function involves plotting a set of points and drawing a smooth curve to create the symmetrical parabola.

The domain of a function consists of all the input values (\(x\)-values) for which the function is defined, while the range is all the possible output values (\(y\)-values). For quadratic functions, and indeed all polynomial functions, the domain is always all real numbers, which is expressed as \((-\infty, \infty)\).

The range, however, depends on the orientation and position of the parabola. For \(f(x) = x^2 + 2\), which opens upwards and has its vertex at the point \((0, 2)\), the smallest \(y\)-value is 2, and it increases without bound as \(x\) moves away from the vertex in either direction. Consequently, the range is given as \([2, \infty)\). Understanding that the domain is not influenced by the orientation or position of the parabola, but the range is significantly determined by its vertex, is essential for graphing.

The vertex of a parabola is a key feature and represents either the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards, respectively. For the function \(f(x) = x^2 + 2\), with its standard form indicating \(ax^2 + c\), we can see that there is no \(bx\) term and thus, the parabola is symmetrically placed over the y-axis.

The vertex is easily found at the point \((0, c)\), with \(c\) being the \(y\)-intercept of the function. In our case, the vertex is at \((0, 2)\), and it's the minimum point since the parabola opens upwards. When graphing the function or analyzing its properties, the vertex serves as a starting point and helps in defining the range of the function as well as in plotting the parabola's symmetry.