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Understanding the domain and range of a quadratic function is essential to sketching and interpreting its graph. The domain represents all possible input values (x-values) for the function. In the function \(y = x^2 - 9\), there are no restrictions on \(x\), meaning \(x\) can be any real number. Therefore, the domain is written as \((-\infty, \infty)\). This is true for most quadratic functions where the denominator is not zero, and there are no square roots or logarithms involved.

The range illustrates all possible output values (y-values). Quadratic functions often have a minimum or maximum point due to their parabolic shape. For a function like \(y = x^2 - 9\), the lowest point on the graph is at \(-9\) since the smallest value of \(x^2\) is \(0\). Hence, the range is all values from \(-9\) to infinity, written as \([-9, \infty)\). This is because as \(x\) increases or decreases without bound, \(y\) becomes larger and larger.

Graph sketching involves understanding the shape and key features of the function. Quadratic functions graph into a shape called a parabola. In the function \(y = x^2 - 9\), the parabola opens upwards, because the \(x^2\) term has a positive coefficient.

Important points on the graph are:

• Vertex: This is the lowest point of the parabola at \((0, -9)\).
• Y-intercept: The graph cuts the y-axis where \(x = 0\), which gives \(y = -9\).
• X-intercepts: These are the points where the graph crosses the x-axis. For \(x = ±3\), \(y = 0\), giving intersection points of \((-3, 0)\) and \((3, 0)\).

With these points plotted, you can draw a smooth curve passing through them, ensuring the parabola opens upwards.

Quadratic functions are polynomial functions of degree two. They take the form \(y = ax^2 + bx + c\). In our example, \(y = x^2 - 9\), the coefficient \(a = 1\), \(b = 0\), and \(c = -9\).

Key aspects of these functions include:

• Parabola: The graph of a quadratic function is always a parabola.
• Symmetry: The parabola is symmetric about its vertex line, known as the axis of symmetry. For our equation, this line is \(x = 0\).
• Direction: The sign of \(a\) determines if the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)). Here, since \(a = 1\), it opens upwards.

These features make quadratic functions predictable and easy to graph once you understand the basic form.