**Find the vertex of the parabola worksheet :**

Here we are going to see some practice questions on vertex of parabola.

(1) Find the vertex of the following parabola

x^{2} = - 4y

(2) Find the vertex of the following parabola

x^{2} − 2x + 8y + 17 = 0

(3) Find the vertex of the following parabola

y^{2} − 8x + 6y + 9 = 0

(4) Find the vertex of the following parabola

x^{2} − 6x − 12y − 3 = 0

**Question 1 :**

Find the vertex of the following parabola

x^{2} = - 4y

**Solution :**

x^{2} = - 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).

**Question 2 :**

Find the vertex of the following parabola

x^{2} − 2x + 8y + 17 = 0

**Solution :**

x^{2} − 2x + 8y + 17 = 0

Subtract 8y and 17 on both sides

x^{2} − 2x + 8y + 17 - 8y - 17 = -8y - 17

x^{2} − 2x = -8y - 17

Split the coefficient of x as the multiple of 2.

x^{2} − 2 **⋅**x **⋅ **1 + 1^{2} - 1^{2} = -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)

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

**Question 3 :**

Find the vertex of the following parabola

y^{2} − 8x + 6y + 9 = 0

**Solution :**

y^{2} + 6y − 8x + 9 = 0

Add 8x and subtract 9 on both sides

y^{2} + 6y − 8x + 9 + 8x - 9 = 0 + 8x - 9

y^{2} + 6y = 8x - 9

Split the coefficient of y as the multiple of 2.

y^{2} + 2**⋅** y **⋅ **3 + 3^{2} - 3^{2} = 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)

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

**Question 4 :**

Find the vertex of the following parabola

x^{2} − 6x − 12y − 3 = 0

**Solution :**

x^{2} − 6x − 12y − 3 = 0

Add 12y and 3 on both sides

x^{2} − 6x − 12y − 3 + 12y + 3 = 0 + 12y + 3

x^{2} − 6x = 12y + 3

Split the coefficient of x as the multiple of 2.

x^{2} − 2**⋅**x**⋅**3 + 3^{2} - 3^{2} = 12y + 3

(x - 3)^{2} - 3^{2} = 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)

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

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