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Understanding the factored form of a quadratic equation is crucial when working with parabolas. The factored form is given by:

• \( y = a(x - r_1)(x - r_2) \)

where \( r_1 \) and \( r_2 \) are the x-intercepts of the parabola. This form is particularly useful because it directly gives you the roots or x-intercepts of the quadratic equation. In this case, we have x-intercepts at \(-3\) and \(9\). Substituting these into the factored form gives us:

• \( y = a(x + 3)(x - 9) \)

Here, \(a\) is a constant that is determined by other conditions, such as another point on the parabola like the vertex. Knowing the x-intercepts helps greatly in graphing and understanding the behavior of the quadratic function.

The vertex form of a quadratic equation is extremely useful for identifying the vertex of a parabola directly. This form is represented as:

• \( y = a(x - h)^2 + k \)

where \((h, k)\) is the vertex. This equation makes it simple to identify the maximum or minimum point of the parabola, known as the vertex. In our exercise, the vertex is given as \((3, -9)\), which means that substituting \(h = 3\) and \(k = -9\), the equation becomes:

• \( y = a(x - 3)^2 - 9 \)

Like in factored form, \(a\) affects the overall shape and direction of the parabola. This form emphasizes the importance of knowing the vertex to graph the parabola accurately.

X-intercepts are the points where the parabola crosses the x-axis of the coordinate plane. These are also known as roots or solutions to the equation. For a quadratic equation given in factored form, the x-intercepts can be seen directly:

• \(x = r_1\) and \(x = r_2\)

where \(r_1\) and \(r_2\) are solutions to the equation when \(y = 0\). This exercise provides the x-intercepts as \(-3\) and \(9\). This tells us that if you replace \(x\) with either \(-3\) or \(9\) in the quadratic equation, the value for \(y\) will be zero. Understanding how to identify the x-intercepts is key to solving quadratics and graphing the solutions. Using these intercepts allows us to write the quadratic equation in factored form without additional computation.

The vertex of a parabola is one of the most significant points on its graph. It represents the turning point and can be either a maximum or minimum point depending on the parabola's orientation. For the vertex form of the equation:

• \( y = a(x - h)^2 + k \)

\( (h, k) \) indicates the vertex. In this exercise, the vertex is given as \( (3, -9) \), which shows where the parabola reaches its lowest point since the parabola opens upwards (because \(a\) is positive). Finding the equation of the parabola involves knowing how this vertex relates to both the factored form and the vertex form. The vertex not only helps in writing the equation but is also crucial in understanding the symmetry of the parabola. Determining the vertex is essential for completely understanding the parabolic shape and can also aid in graphing or solving real-world problems involving quadratics.