Determine the equation of the parabola

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In summary, a parabola is a symmetrical, U-shaped curve formed by the intersection of a plane and a cone. Its equation is in the form of y = ax^2 + bx + c and can be determined by knowing the coordinates of at least three points on the curve. The values of a, b, and c in the equation determine the shape and position of the parabola. The equation can also be determined using the focus and directrix. To graph a parabola, you can use the vertex form of the equation or the y-intercept and x-intercepts.
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Determine the equation of the parabola with range y|y≧-6 and x-intercepts at -5 and 3.
 
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$y = k(x+5)(x-3)$, where $k$ is a constant

$y \ge -6 \implies k > 0$

vertex is located at $(x,-6)$ where $x$ is midway between the two x-intercepts

determine the x-value of the vertex, then use the vertex coordinates to determine the value of $k$
 

Related to Determine the equation of the parabola

What is a parabola?

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.

How do you determine the equation of a parabola?

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.

What is the significance of the coefficient a in the equation of a 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.

Can the equation of a parabola be written in different forms?

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.

What are some real-life applications of parabolas?

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.

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