parabola equation

The general equation of a parabola is: y = a (x-h) 2 + k or x = a (y-k) 2 +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y 2 = 4ax. Some of the important terms below are helpful to understand the features and parts of a parabola y 2 = 4ax. Focus: The point (a, 0) is the focus of the parabola the focus. F = ( − b 2 a , 4 a c − b 2 + 1 4 a ) {\displaystyle F=\left (- {\frac {b} {2a}}, {\frac {4ac-b^ {2}+1} {4a}}\right)} , the directrix. y = 4 a c − b 2 − 1 4 a {\displaystyle y= {\frac {4ac-b^ {2}-1} {4a}}} , the point of the parabola intersecting the y axis has coordinates. ( 0 , c ) {\displaystyle (0,c)} Factor out \(4 p\) and we have the standard equation for a parabola: \[(x-h)^{2}=4 p(y-k) \] This equation will be different depending on the orientation of the parabola. An upward facing parabola will have this standard equation and both sides will have the same sign. The simplest equation for a parabola is y = x2 Turned on its side it becomes y2 = x (or y = √x for just the top half) A little more generally: y 2 = 4ax where a is the distance from the origin to the focus (and also from the origin to directrix) General Equations of Parabola. The general equation of a parabola is given by y = a(x – h) 2 + k or x = a(y – k) 2 +h. Here, (h, k) denotes the vertex. y = a(x – h) 2 + k is the regular form. x = a(y – k) 2 +h is the sidewise form. Position of a point with respect to the parabola The parabola equation is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis and y-axis. The four such possible orientations of the parabola are explained in the table below: Thus, we can derive the equations of the parabolas as: y 2 = 4ax y 2 = -4ax x 2 = 4ay x 2 = -4ay Let’s take a look at the first form of the parabola. f (x) = a(x −h)2 +k f ( x) = a ( x − h) 2 + k. There are two pieces of information about the parabola that we can instantly get from this function. First, if a a is positive then the parabola will open up and if a a is negative then the parabola will open down. Free Parabola calculator - Calculate parabola foci, vertices, axis and directrix step-by-step If we start at the vertex (it does not matter where it is on the graph), go over 1 and count how much you go up or down to determine the magnitude. Several examples and for simplicity's sake, keep the vertex at the origin. If I go over one up two, then the equation is y = 2x^2. over 1 up 3 it is y = 3x^2, over 1 down 1, then y = - x^2, over 1 ... A parabola is the set of all points ( x, y) in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. The standard form of a parabola with vertex ( 0, 0) and the x -axis as its axis of symmetry can be used to graph the parabola. The graph of the quadratic function is a U-shaped curve is called a parabola. The graph of the equation y = x 2, shown below, is a parabola. (Note that this is a quadratic function in standard form with a = 1 and b = c = 0.) In the graph, the highest or lowest point of a parabola is the vertex. The vertex of the graph of y = x 2 is (0, 0). Conic Form of Parabola Equation: (x - h)2 = 4p(y - k) with the vertex at (h, k), the focus. at (h, k+p) and the directrix. y = k - p. Since the example at the right is a translation of the previous graph, the relationship between the parabola and its focus and directrix remains the same (p = ¼). A parabola is a symmetrical, curved, U-shaped graph. The equation of a parabola graph is y = x². Parabolas exist in everyday situations, such as the path of an object in the air, headlight shapes ...