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Azurin
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Determine the equation of the parabola with range y|y≧-6 and x-intercepts at -5 and 3.
A parabola is a type of curve that is commonly seen in mathematics and physics. It is a symmetrical curve that is shaped like a U or an upside-down U. It is formed by the graph of a quadratic function, which is a polynomial function of the form y = ax^2 + bx + c.
The equation of a parabola can be determined by using the coordinates of three points on the curve. These points can be either the vertex, the focus, and a point on the parabola, or three points that are equidistant from each other. By plugging in the coordinates of these points into the general form of a quadratic equation, the values of a, b, and c can be solved for, giving the equation of the parabola.
The coefficient a in the equation of a parabola determines the shape and direction of the curve. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. The absolute value of a also affects the steepness of the curve, with a larger absolute value resulting in a steeper curve.
Yes, the equation of a parabola can be written in different forms, such as vertex form, standard form, and intercept form. These forms are useful for different purposes, such as finding the vertex, the focus, or the x- and y-intercepts of the parabola. However, they all represent the same curve and can be converted into each other.
Parabolas have many real-life applications, including in physics, engineering, and architecture. They can be used to model the trajectory of a projectile, the shape of a satellite dish, and the design of a suspension bridge. They are also commonly seen in art and design, such as in the shape of a fountain or a roller coaster.