# FIND THE VERTEX OF THE PARABOLA

## About "Find the vertex of the parabola"

Find the vertex of the parabola :

By comparing the given with any one of the following standard form, we can easily find the vertex of the parabola.

 Equation Axis of symmetry Which side open (y-k)² = 4a(x-h)(y-k)² = -4a(x-h)(x-h)² = 4a(y-k)(x-h)² = -4a(y-k) x-axisx-axisy-axisy-axis Right wardLeft wardUpwardDownward

In the above equations (h, k) is the vertex of the parabola.

Let us look in to some example problems based on the above concept.

Example 1 :

Find the vertex of the following parabola

x2 = - 4y

Solution :

x2 = - 4y

We can compare the above equation with the general form (x - h)2  = -4 a (y - k)

(x - 0)2 = - 4(y - 0)

Hence, the required vertex V(h, k) is (0, 0).

Example 2 :

Find the vertex of the following parabola

x2 − 2x + 8y + 17 = 0

Solution :

x2 − 2x + 8y + 17 = 0

Subtract 8y and 17 on both sides

x2 − 2x + 8y + 17 - 8y - 17  =  -8y - 17

x2 − 2x  =  -8y - 17

Split the coefficient of x as the multiple of 2.

x2 − 2 1 + 12 - 12   =  -8y - 17

(x - 1)2 - 1   =  -8y - 17

(x - 1)2 - 1 + 1   =  -8y - 17 + 1

(x - 1)2   =  -8y - 16

(x - 1)2   =  -8(y + 2)

(x - 1)2   =  -8(y - (-2))

The above equation exactly matches with the equation

(x - h)2   =  -4a(y - k)

(h, k)  ==>  (1, -2)

Hence, the required vertex of the parabola is (1, -2).

Example 3 :

Find the vertex of the following parabola

y2 − 8x + 6y + 9 = 0

Solution :

y2 + 6y − 8x + 9  =  0

Add 8x and subtract 9 on both sides

y2 + 6y − 8x + 9 + 8x - 9  =  0 + 8x - 9

y2 + 6y  =  8x - 9

Split the coefficient of y as the multiple of 2.

y2 + 2 y 3 + 32 - 32  =  8x - 9

(y - 3)2 - 9  =  8x - 9

(y - 3)2 - 9 + 9  =  8x - 9 + 9

(y - 3)2  =  8x

(y - k)2  =  4a (x - h)

(y - (-3))2  =  8(x - 0)

(h, k)  ==>  (0, -3)

Hence, the required vertex of the parabola is (0, -3).

Example 4 :

Find the vertex of the following parabola

x2 − 6x − 12y − 3 = 0

Solution :

x2 − 6x − 12y − 3 = 0

Add 12y and 3 on both sides

x2 − 6x − 12y − 3 + 12y + 3 = 0 + 12y + 3

x2 − 6x  = 12y + 3

Split the coefficient of x as the multiple of 2.

x2 − 2x3 + 32 - 32 = 12y + 3

(x - 3)2 - 32 = 12y + 3

(x - 3)2 - 9 = 12y + 3

(x - 3)2 - 9 + 9 = 12y + 3 + 9

(x - 3)2  = 12y + 12

(x - 3)2 = 12(y + 1)

(x - 3)2 = 12(y - (-1))

(x - h)2  =  4a (y - k)

(x - 3)2  =  12(y - (-1))

(h, k)  ==>  (3, -1)

Hence, the required vertex of the parabola is (3, -1).

After having gone through the stuff given above, we hope that the students would have understood "Find the vertex of the parabola".

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