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Parabola : In algebra, dealing with parabolas usually means graphing quadratics or finding the max/min points (that is, the vertices) of parabolas for quadratic word problems. In the context of conics, however, there are some additional considerations.
There are different types of equations in which, we get a parabola.
The standard form of these equations with vertex as origin are :
1) y2 = 4ax for a >0. The parabola is, therefore, symmetrical about x-axis, which is the axis of symmetry of parabola.
2) y2 = -4ax for a < 0, x may have negative value or zero but no positive value. Therefore, in this case the parabola opens to left. The axis of symmetry is again the x-axis.
3) x2 = 4ay for a > 0. In this case parabola opens upward. The axis of symmetry is the y-axis.
4) y2 = -4ay for a < 0.The parabola opens downward and the axis of symmetry is again the y-axis.

1) Quadratic equation - > f(x)= y = ax2 + bx + c
Example :The quadratic equation as f(x) = x2+ 5x + 6
Factors of the given equation are (x +3)(x +2)
∴ x- intercepts are x = -3 and x = -2. Mark these points on the x- axis.
Vertex : X coordinate of vertex = -b / 2a
From the given equation b= 5 and a = 1
x = -5/2(1) = -5/2 = -2.5
To find the y-coordinate of vertex put x = -2.5 in the given equation
y = (- 2.5)2 + 5(-2.5) + 6
y = 6.25 - 12.5 + 6
y = 12.25 - 12.5
y = - 0.25
Coordinates of vertex = ( -2.5,-0.5)
Mark the coordinates of vertex in the graph.
Now, join x-intercepts and vertex as a curve. This curve is a Parabola.

2) When vertex as origin there are 4 types of parabola.
The standard form is y2 = 4ax
Graph Dictionary