Derivation of Vertex of a Parabola Formula
Suppose we have a parabola with standard equation as, y = ax2 + bx + c.
This can be written as,
y – c = ax2 + bx
y – c = a (x2 + bx/a)
Adding and subtracting b2/4a2 on the RHS, we get
y – c = a (x2 + bx/a + b2/4a2 – b2/4a2)
y – c = a ((x + b/2a)2 – b2/4a2)
y – c = a (x + b/2a)2 – b2/4a
y = a (x + b/2a)2 – b2/4a + c
y = a (x + b/2a)2 – (b2/4a – c)
y = a (x + b/2a)2 – (b2 – 4ac)/4a
We know, D = b2 – 4ac, so the equation becomes,
y = a (x + b/2a)2 – D/4a
Comparing the above equation with the vertex form y = a(x – h)2 + k, we get
h = -b/2a and k = -D/4a
This derives the formula for coordinates of the vertex of a parabola.
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Vertex of a Parabola Formula
Vertex of a Parabola Formula: The point where the parabola and its axis of symmetry intersect is called the vertex of a parabola. It is used to determine the coordinates of the point on the parabola’s axis of symmetry where it crosses it. For the standard equation of a parabola y = ax2 + bx + c, the vertex point is the coordinate (h, k). If the coefficient of x2 in the equation is positive (a > 0), then the vertex lies at the bottom else it lies on the upper side.
In this article, we will discuss the vertex of a parabola, its formula, derivation of the formula, and solved examples on it.
Table of Content
- Properties of Vertex of a Parabola
- Vertex of a Parabola Formula
- Derivation of Vertex of a Parabola Formula
- Sample Problems on Vertex of a Parabola Formula
Vertex of a Parabola