Finding the equation of a parabola

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In summary, to find the equation of a parabola given its vertex and 1 point, you can use the general quadratic equation y=ax^2+bx+c and its derivative, y'=2ax+b. By setting the derivative to zero at the vertex, you can solve for the value of a and then plug it into the equation. Alternatively, if the parabola has a vertical line of symmetry, you can use the specific formula y = a(x-x0)^2 + y0, where x0 and y0 are the coordinates of the vertex, and solve for a using the given point. Graphing the parabola can also provide a shortcut for finding the equation.
  • #1
Flatland
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How do you find the equation of a parabola if you are given it's vertex and 1 point? For example, find the quadratic equation of a parabola that has a vertex of (-2,-2) and goes through the point (-1,0)
 
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General quadratic equation: y=ax^2 + bx + c

Derivative: y' = 2ax + b

At the vertex, the derivate equals to zero. Use this fact and simultaneous equations to arrive at the equation.
 
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  • #3
More simply, since you are given the vertex of the parabola, you can write the equation y= a(x-x0)2+ y0 where x0 and y0 are the coordinates of the vertex. Choose a to force the parabola to go through the second point.

y= a(x-(-2))2- 2= a(x+2)2- 2. Setting x= -1, y= 0,
0= a(-1+2)2-2= a- 2 so a= 2.

By the way, this is assuming the parabola has a vertical line of symmetry. Otherwise there are an infinite number of parabolas satisfying these conditions.
 
  • #4
Know your parabolas

HallsofIvy said:
... assuming the parabola has a vertical line of symmetry...
Graph first, and you may find a shortcut for a given specific data.
If a point on a parabola is 1 to the right and 2 up from its vertex, it must be parabola
[tex]y = 2x^{2}[/tex]
shifted horizontally and vertically, so its vertex (0,0) moves into (-2,-2), i.e. 2 to the left and 2 down:
[tex]y = 2(x+2)^{2} - 2[/tex]
 

What is a parabola?

A parabola is a symmetrical arc-shaped curve that is formed by the intersection of a plane and a cone. It is a type of conic section and can be seen in many real-life objects such as bridges, satellite dishes, and water fountains.

What is the general equation of a parabola?

The general equation of a parabola is y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The value of a determines the shape and direction of the parabola, while b and c affect its position on the coordinate plane.

How do you find the equation of a parabola given three points?

To find the equation of a parabola given three points, you can use the method of substitution. First, substitute the coordinates of the three points into the general equation of a parabola. Then, you can solve the resulting system of equations to find the values of a, b, and c, and thus the equation of the parabola.

What is the focus of a parabola?

The focus of a parabola is a fixed point that lies on the axis of symmetry and is equidistant from all points on the curve. It is also the point where all reflected light rays converge if the parabola is used as a reflective surface.

What is the directrix of a parabola?

The directrix of a parabola is a straight line that is perpendicular to the axis of symmetry and is equidistant from all points on the curve. It is used to define the shape and position of the parabola and can be used to find the focus and vertex of the parabola.

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