**Vertex Form of a Quadratic Equation :**

**Learning Objectives : **

* Vertex form of a quadratic equation.

* If a quadratic equation is given in standard form, how to write it in vertex form.

* How to sketch the graph of a quadratic equation that is in vertex form.

The vertex form of a quadratic equation is given by

**y = a(x - h) ^{2} + k**

where (h, k) is the vertex of the parabola.

The h represents the horizontal shift and k represents the vertical shift.

**Horizontal Shift :**

How far left or right the graph is shifted from x = 0 from the parent equation y = x^{2}.

**Vertical Shift :**

How far up or down the graph is shifted from y = 0 from the parent equation y = x^{2}.

**Example :**

In the graph (Parabola) of a quadratic equation shown above, the graph is shifted 2 units to the right from x = 0 and 1 unit up from y = 0.

So, the vertex is

(Horizontal shift, Vertical shift) = (2, 1)

**Example :**

Write the quadratic equation in vertex form and write its vertex :

y = - x^{2} + 2x - 2

**Solution :**

**Vertex form : **

y = - x^{2} + 2x - 2

y = - (x^{2} - 2x + 2)

y = - (x^{2} - 2 ⋅ x ⋅ 1 + 2)

y = - (x^{2} - 2 ⋅ x ⋅ 1 + 1^{2 }- 1^{2 }+ 2)

y = - [(x - 1)^{2 }- 1^{2 }+ 2]

y = - [(x - 1)^{2 }- 1^{ }+ 2]

y = - [(x - 1)^{2 }+ 1]

y = - (x - 1)^{2} - 1

**Vertex : **

Comparing the equations

y = a(x - h)^{2} + k and y = - (x - 1)^{2} - 1,

we get

(h, k) = (1, -1)

So, the vertex is (1, -1).

**Example :**

Write the quadratic equation in vertex form and sketch its graph :

y = - x^{2} - 2x + 3

**Solution :**

**Vertex form : **

y = - x^{2} - 2x + 3

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

y = - (x^{2} - 2 ⋅ x ⋅ 1 - 3)

y = - (x^{2} - 2 ⋅ x ⋅ 1 + 1^{2} - 1^{2} - 3)

y = - [(x - 1)^{2 }- 1^{2} - 3]

y = - [(x - 1)^{2 }- 1 - 3]

y = - [(x - 1)^{2 }- 4]

y = - (x - 1)^{2} + 4

**Vertex : **

Comparing the equations

y = a(x - h)^{2} + k and y = - (x - 1)^{2} + 4,

we get

(h, k) = (1, 4)

So, the vertex is (1, 4).

**Graph :**

To graph the above quadratic equation, we need to find x-intercepts and y-intercepts, if any.

**x-intercept : **

To find the x-intercept, put y = 0.

0 = - (x - 1)^{2} + 4

(x - 1)^{2} = 4

Take radical on each side.

x - 1 = ±√4

x - 1 = ± 2

x - 1 = - 2 or x - 1 = 2

x = - 1 or x = 3

**So, the x-intercepts are -1 and 3 . **

**y-intercept : **

To find the y-intercept, put x = 0.

y = - (0 - 1)^{2} + 4

y = - (-1)^{2} + 4

y = - 1 + 4

y = 3

**So, the y-intercept is 3.**

In the vertex form of the given quadratic equation, we have negative sign in front of (x - 1)^{2}. So, the graph of the given quadratic equation will be open downward parabola.

After having gone through the stuff given above, we hope that the students would have understood, "Vertex Form of a Quadratic Equation".

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