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Let's first discuss the vertex form of a quadratic function. The vertex form of a quadratic function is written as \(y = a(x-h)^2 + k\). Here, \(a\) determines the direction and width of the parabola. The values \(h\) and \(k\) give us the coordinates of the vertex of the parabola, making it easier to graph. The vertex form is very useful because it tells us directly where the vertex is, allowing us to easily draw the parabola. In our problem, the function is \((x-1)^2 - 3\), which is already in vertex form. Here, \(h = 1\) and \(k = -3\), so the vertex is at \((1, -3)\).

Next, we need to find the axis of symmetry for the given quadratic function. For any quadratic function in vertex form \(y = a(x-h)^2 + k\), the axis of symmetry is a vertical line that goes through the vertex. This line helps us understand that both sides of the parabola mirror each other. The equation for the axis of symmetry is given by \(x = h\). In this case, since \(h = 1\), the axis of symmetry is the vertical line \(x = 1\).

Let's move on to determining the domain and range of the given quadratic function. The domain of any quadratic function is all real numbers because there are no restrictions on the values that \(x\) can take. Thus, the domain is \((-\infty, \infty)\). The range, on the other hand, depends on the direction in which the parabola opens. Since the coefficient of \((x-1)^2\) is positive, the parabola opens upwards. This means the minimum point occurs at the vertex. Therefore, the range is all values greater than or equal to the y-coordinate of the vertex, which is \(-3\). So the range is \[y \geq -3\ \text{or} \ [-3, \infty)\].

Finally, let's discuss the general concept of a quadratic function. A quadratic function is a polynomial function of degree 2, generally written as \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants and \(a \eq 0\). It graphs into a U-shaped curve known as a parabola. The parabola can either open upwards or downwards, depending on the sign of \(a\). For positive \(a\), the parabola opens upwards, and for negative \(a\), it opens downwards. Quadratic functions are used in various fields, including physics, engineering, economics, and statistics, to model different types of phenomena. Understanding its vertex form, axis of symmetry, and domain and range helps in analyzing and graphing the function more effectively.