**How to find the vertex of parabola :**

The vertex of a parabola is the point where the parabola crosses its axis of symmetry.

The vertex of the parabola is the highest or lowest point also known as maximum value or minimum value of the parabola.Here we are going to see how to find the vertex of the parabola.

If the parabola is open upward then vertex will be the lowest point

If the parabola is open downward then the vertex will be the highest point of the parabola.

The standard equation of the parabola is

**y = ax² + bx + c**

Vertex form of the parabola :

**y = a (x - h)² + k**

here (h, k) is known as vertex of the parabola.

To find the value of "h" we need to use the formula "-b/2a". By applying this x -coordinate in the given equation we will get the value of y that is "k".

Let us see some example problems to understand the method.

**Example 1 :**

Find the vertex of the parabola y = x² - 2 x - 5

**Solution :**

**Method 1 :**

Compare the given equation with the general form of quadratic equation

y = x² - 2 x - 5

y = ax² + b x + c

a = 1, b = -2 and c = -5

x - coordinate of vertex = -b/2a ==> - (-2)/2(1) ==> 1

Now we have t o apply x = 1 in the given equation according to get the value of y.

y = (1)² - 2(1) - 5

y = 1 - 2 - 5

y = 1 - 7 ==> -6

Vertex of the parabola is (1,6)

We can do the same problem in another method also.

**Method 2 :**

y = x² - 2 x - 5

Now we are going to convert the given equation in vertex form using completing the square method.

y = x² - 2 x(1) + 1² - 1² - 5

y = (x - 1) - 6

y = a (x - h)² + k

(h, k) ==> (1, 6)

Hence the vertex of the parabola is (1, 6).

Let us see the next example problem on "Find the vertex of parabola".

**Example 2 :**

Find the vertex of the parabola y = -x² - 14 x - 59

**Solution :**

**Method 1 :**

Compare the given equation with the general form of quadratic equation

y = -x² - 14 x - 59

y = ax² + b x + c

a = -1, b = -14 and c = -59

x - coordinate of vertex = -b/2a ==> - (-14)/2(-1) ==> -7

Now we have t o apply x = 7 in the given equation according to get the value of y.

y = -(-7)² - 14(-7) - 59

y = -49 + 98 - 59

y = -108 + 98 ==> -10

Vertex of the parabola is (-7, -10)

We can do the same problem in another method also.

**Method 2 :**

y = -x² - 14 x - 59

y = -[x² + 14 x + 59]

Now we are going to convert the given equation in vertex form using completing the square method.

y = -[x² + 2 x(7) + 7² - 7² + 59]

y = -[(x + 7)² - 49 + 59]

y = -[(x + 7)² + 10]

y = a (x - h)² + k

(h, k) ==> (-7, -10)

Hence the vertex of the parabola is (-7, -10).

Let us see the next example problem on "Find the vertex of parabola".

**Example 3 :**

Find the vertex of the parabola y = x² + 4 x

**Solution :**

**Method 1 :**

Compare the given equation with the general form of quadratic equation

y = x² + 4 x

y = ax² + b x + c

a = 1, b = 4 and c = 0

x - coordinate of vertex = -b/2a ==> -4/2(1) ==> -2

Now we have t o apply x = -2 in the given equation according to get the value of y.

y = (-2)² + 4 (-2)

y = 4 - 8 ==> -4

Vertex of the parabola is (-2, -4)

We can do the same problem in another method also.

**Method 2 :**

y = x² + 4 x

Now we are going to convert the given equation in vertex form using completing the square method.

y = x² + 2 x(2) + 2² - 2²

y = (x + 2)² - 4

(h, k) ==> (-2, -4)

Hence the vertex of the parabola is (-2, -4).

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

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