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-axis

x-axis

y-axis

y-axis

Right ward

Left ward

Upward

Downward

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)

So, 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

Add 1 on both sides

(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)

So, 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

Add 9 on both sides

(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)

So, 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

Add 9 on both sides

(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)

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

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