Vertex Formula: Introduction, Derivation & Solved Examples

Vertex of the parabola is the intersection point of the parabola and the axis of symmetry. In simpler words, the point at which the parabola crosses the axis of symmetry is called the vertex. Depending on the vertex, the parabola changes its direction. If the vertex is maximum, the parabola opens down. If the vertex is minimum, the parabola opens up. The standard form of the parabola is y=ax²+bx+c. If a > 1, then the parabola is an upward parabola. If a < 1, then the parabola is a downward parabola. As per the equation, the symmetry axis is along the y-axis. If the equation has a y² term, then the symmetry axis is along the x-axis. The general vertex form of the parabola is y = a(x-h)² + k.

Key Terms: Vertex Formula, Parabola, Coordinates, symmetry acis, vertex, derivation of vertex formula

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Introduction to Vertex Formula

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The vertex formula is used in order to find the vertex of a parabola. The vertex form of the parabola is given below.

y=a(x-h)²+k

Here, h, k are vertex coordinates.

The vertex (h,k) can be determined in two different ways. 

• (h,k) = \(\frac{-b}{2a}\), \(\frac{-D}{4a}\)

Here, D is the discriminant which can be determined by D(discriminant)=b²-4ac

h and k are the vertex coordinates

• h = \(\frac{-b}{2a}\)

Here, h is the x-coordinate of the vertex. Later, h can be used to find k by evaluating it in y.

Vertex Formula

Discover about the Chapter video:

Conic Sections Detailed Video Explanation:

Also Read: Parabola Formula

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Vertex Formula Derivation

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Let us derive the first formula y = a(x-h)² + k

The standard form of the parabola is 

y=ax² + bx + c ...(1)

Subtracting c on both sides,

y – c = ax² + bx ...(2)

Consider a as common factor, 

y – c = a(x²+\(\frac{bx}{a}\)) ...(3)

Here, half the coefficient of x is b/2a. Its square is b²/4a².

Add and subtract b²/4a² on the right side of equation (3)

y – c = a( x² + \(\frac{bx}{a}\) + \(\frac{b^2}{4a^2}\) – \(\frac{b^2}{4a^2}\) ) ...(4)

Here, x² + \(\frac{bx}{a}\) + \(\frac{b^2}{4a^2}\)  can be written as (x+\(\frac{b}{2a}\))².

Equation (4) becomes

y – c = a((x + \(\frac{b}{2a}\))² – \(\frac{b^2}{4a^2}\) )...(5)

Distribute a on the right side, 

y – c = a(x + \(\frac{b}{2a}\))² – \(\frac{b^2}{4a}\)  ...(6)

Add c on both sides, 

y = a(x + \(\frac{b}{2a}\))² – \(\frac{b^2}{4a}\)  + c ...(7)

Write the equation (7) in the given form

y = a(x + \(\frac{b}{2a}\))² – \(\frac{b^2-4ac}{4a}\)   ..(8)

Comparing equation (8) with parabola equation y = a(x-h)² + k

We get, 

h = – b/2a

k =  – \(\frac{b^2-4ac}{4a}\)

We know that D(discriminant) =  b² – 4ac.

Therefore, the vertex formula can be written as 

(h,k) = (\(\frac{-b}{2a}\), \(\frac{-D}{4a}\))

Vertex of a Parabola

Also Read: Parabola

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Things to Remember

• Vertex of the parabola is the intersection point of the parabola and the axis of symmetry. In simpler words, the point at which the parabola crosses the axis of symmetry is called the vertex. 
• Depending on the vertex, the parabola changes its direction. If the vertex is maximum, the parabola opens down. If the vertex is minimum, the parabola opens up.
• The vertex form of the parabola is y=a(x-h)²+k. Here, h, k are vertex coordinates.
• When the symmetry axis is along the x-axis, the parabola opens (a) right if x-coefficient is positive and (b) left if x-coefficient is negative. When the symmetry axis is along the y-axis, the parabola opens (a) upward if the y-coefficient is positive and (b) downward if the y-coefficient is negative.
• The vertex (h,k) can be determined in two different ways. 
  • (h,k) = (\(\frac{-b}{2a}\), \(\frac{-D}{4a}\)) Here, D is the discriminant and h and k are the vertex coordinates.
  • h=\(\frac{-b}{2a}\), Here, h is the x-coordinate of the vertex. Later, h can be used to find k by evaluating it in y.