WRITE THE QUADRATIC EQUATION IN VERTEX FORM

The vertex form of a quadratic function is

If a > 0, then the parabola opens upward.

If a < 0, then the parabola opens downward.

Write the following quadratics in vertex form  by completing the square and find the vertex.

Example 1 :

y  =  x2 + 2x + 4

Solution :

Using completing the square method,

y  =  x2 + 2x⋅1 + 12 -1+ 4

y  =  (x+1)2 -1+ 4

y  =  (x+1)2 + 3

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (-1, 3)

Example 2 :

y = x2-6x + 3

Solution :

Using completing the square method,

y  =  x2 - 2x⋅3 + 32 - 3+ 3

y  =  (x-3)2 -9 + 3

y  =  (x-3)2 - 6

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (3, -6)

Example 3 :

y  =  x2 - 2x

Solution :

Using completing the square method,

y  =  x2 - 2x⋅1 + 12 - 12

y  =  (x-1)2 -1

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (1, -1)

Example 4 :

y  =  x2 - 3x + 1

Solution :

Using completing the square method,

y  =  x2 - 2x⋅(3/2) + (3/2)2 - (3/2)2

y  =  (x-(3/2))2 - (9/4)

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (3/2, -9/4)

Example 5 :

y  =  -x2 + 2x + 8

Solution :

By factoring negative from the quadratic function, we get

y  =  -(x2 - 2x - 8)

y  =  -(x2 - 2x - 8)

y  =  -(x2 - 2x⋅1 + 12-1-8)

y  =  -((x-1)2-9)

y  =  -(x-1)2+9

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (1, 9)

Example 6 :

y  =  (x-1)(x-3)

Solution :

y  =  x2-4x+3

y  =  x2-2x⋅2+22-22+3

y  =  (x-2)2-4+3

y  =  (x-2)2-1

By comparing this with the vertex form of parabola, we get

(h, k)  ==>  (2, -1)

Example 7 :

y  =  3(x+1)2

Solution :

y  =  3(x+1)2 + 0

(h, k)  ==> (-1, 0)

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