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Understanding the domain and range of a function is critical in the study of mathematics, especially when graphing parabolas. The domain refers to the set of all possible input values for the function, which are the x-values on the coordinate axes. In the case of quadratic functions, such as \(f(x)=x^2+1\) and \(g(x)=x^2-1\), the domain is typically all real numbers because you can input any real number and get a corresponding output. Mathematically, we express this as \((-\infty, \infty)\).

The range, on the other hand, encompasses all possible output values or y-values that a function can produce. For upward opening parabolas represented by quadratic functions, the range is determined by the vertex, which is the highest or lowest point on the graph, depending on the direction the parabola opens. Since both parabolas \(f(x)\) and \(g(x)\) open upwards in our example, their ranges start at their respective vertices and extend to infinity. This gives us ranges of \([1, \infty)\) for \(f(x)\) and \([-1, \infty)\) for \(g(x)\).

It's helpful for students to remember that the domain and range are essential when describing the behavior of functions and ensuring that all potential values are considered.

The coordinate axes are the foundation for graphing any function, including parabolas formed by quadratic functions. They consist of the horizontal x-axis and the vertical y-axis, which intersect at the origin, denoted as point (0,0). When graphing a quadratic function like \(f(x)=x^2+1\) or \(g(x)=x^2-1\), students starts by identifying the vertex of the parabola, which lies on the y-axis for these particular functions because their vertex has an x-coordinate of 0.

Once the vertex is plotted on the coordinate axes, the shape of the parabola is drawn by plotting additional points and using the symmetry about the vertical line that passes through the vertex, known as the axis of symmetry. This axis is always a vertical line where the x-value is constant; for our example, the axis of symmetry for both parabolas would be the y-axis itself. Recognizing the role of the coordinate axes helps students not only in plotting the graph correctly but also in understanding how the function behaves in the coordinate plane.

A quadratic function is a type of polynomial function where the highest power of the variable x is two. Its general form is \(ax^2+bx+c\), where a, b, and c are constants and the value of a is not zero. In our exercise, the functions \(f(x)=x^2+1\) and \(g(x)=x^2-1\) are examples of quadratic functions. These functions produce graphs known as parabolas.

The parabolas in this exercise open upwards, as indicated by the positive coefficient of the x-squared term. This means that as x-values become larger in magnitude—both positively and negatively—the y-values produced by the function also become larger. Understanding how the coefficients in the quadratic function affect the graph's shape and position is fundamental for students. For instance, the constant term (1 in \(f(x)\) and -1 in \(g(x)\)) determines the y-intercept and the initial value of the range. A key point for students to remember is that the vertex of the parabola lies on the axis of symmetry and significantly affects the range of the function.

In summary, quadratic functions generate parabolas on the coordinate plane, and recognizing their standard components such as vertex, axis of symmetry, and the effect of each term can greatly aid in graphing and understanding their behavior.