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Parabolas are U-shaped curves that can open either upwards or downwards. They are the graphs of quadratic functions of the form \(y = ax^2 + bx + c\). Key features of a parabola include its vertex, axis of symmetry, and direction of opening. The vertex is the highest or lowest point on the parabola. The axis of symmetry is a vertical line that runs through the vertex, dividing the parabola into two mirror-image halves. The direction of opening depends on the coefficient of \(x^2\): if it’s positive, the parabola opens upwards; if negative, it opens downwards. For instance, the given equations \(y = 7x^2 - 5x - 9\) and \(y = -2x^2 + 4x + 9\) lead to parabolas opening upwards and downwards, respectively.

Intersection points are where two or more graphs meet. To find these points for the parabolas given by \(y = 7x^2 - 5x - 9\) and \(y = -2x^2 + 4x + 9\), we set the equations equal to each other: \(7x^2 - 5x - 9 = -2x^2 + 4x + 9\).
Combining and simplifying this equation leads to another quadratic \(x^2 - x - 2 = 0\).
By factoring, we find the x-values where the parabolas intersect: \(x = 2\) and \(x = -1\).
Substituting these x-values back into either original equation gives us the y-values of the intersection points: \((2, 9)\) and \((-1, 3)\).

Graphing functions involves plotting their equations on a coordinate plane to visualize their behavior and how they interact with other functions. Start by identifying key features such as vertex, axis of symmetry, and intercepts. For our functions \(y = 7x^2 - 5x - 9\) and \(y = -2x^2 + 4x + 9\), plot several points on each curve, including the vertex, and draw a smooth curve through these points. Next, mark the intersection points \( (2, 9) \) and \( (-1, 3) \) on the graph to show where the parabolas intersect. This visual representation helps in understanding the nature of the functions and their interactions.

Factoring quadratic equations is a crucial skill in solving them. A quadratic equation is generally in the form \(ax^2 + bx + c = 0\). To solve it, we look for two numbers that multiply to give \(ac\) and add to give \(b\).
As in our solution, \(x^2 - x - 2 = 0\) can be factored into \((x - 2)(x + 1)\).
Setting each factor to zero gives the solutions \(x = 2\) and \(x = -1\).
This method is effective when we can easily identify the factors. In cases where factoring is not straightforward, completing the square or using the quadratic formula may be necessary. Factoring simplifies the process and helps in understanding the roots of the quadratic equation.