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How to convert Standard form to Vertex form

Quadratic equation of parabola: Standard to Vertex form

The equation of the parabola in standard form:

y = ax2 + bx + c 

Where, a, b, c are constants and real numbers, and a ≠ 0.

x and y are variables, where (x,y) represents a point on the parabola. 

The equation of the parabola in vertex form:

y = a(x-h)2 + k

Where, a, h, k are constants and real numbers, and a ≠ 0.

x and y are variables, where (x,y) represents a point on the parabola. 

From standard form to vertex form conversion:

Our sample parabola to solve:

y= 5x2 + 10x + 10 

(You can write the equations on the Graph plotter to see how the parabola looks like).

Step 1: Make sure the coefficient of x2 is 1; if not separate the number as a common factor.

First, we will have to make sure that the coefficient of x2 is 1 (i.e. the value of a). If the coefficient is not 1 we will have to separate the number outside as a common factor. 

Now in our sample equation, the coefficient is not 1, it is 5. So we will have to separate 5 outside as a common number. 

y = 5x2 + 10x + 10

⇒ y = 5 (x2 + 2x + 2)

Now the coefficient of x2 is 1.

Step 2: Identify the coefficient of x.

Identify the value of b or the coefficient of x. It is 2.

Step 3: Divide the coefficient of x by 2, and square the resultant number.

Coefficient of x ➗ 2 = 2➗2 = 1

So the resultant number is 1.

(The resultant number)2 = (1)2 

Step 4: Add the above square number on both sides of the equation.

Step 5: Factorize the perfect square trinomial.

y + 12 = 5 (x2 + 2x + 2) + 12 

⇒ y + 12 = 5 (x2 + 2x + 1 + 1) + 12 

⇒ y + 12 = 5 {(x2 + 2x + 1) + 1} + 12 

⇒ y + 12 = 5 {(x + 1)2 + 1} + 12 

Step 6: Simplify the other numbers. The resultant equation is the vertex form. 

 y + 12 = 5 {(x + 1)2 + 1} + 12 

⇒ y + 12 = 5 {(x + 1)2 + 1} + 12 

⇒ y + 1 = 5 (x + 1)2 + 5 + 1 

⇒ y = 5 (x + 1)2 + 5 + 1 – 1

⇒ y = 5 (x + 1)2 + 5 

                             This is the vertex form.

So the vertex is -1, 5.

We can verify the answer with the completing square method equation: 

x = – b / 2a

Here, a = 5 , b = 10 (comparing y= 5x2 + 10x + 10 and y = ax2 + bx + c).

So, putting the values in the equation,

⇒ x = –  10/ (2 X 5)

⇒ x = –  10/ 10

⇒ x = –  1.

Now, If we put the value x = -1 on the standard form equation, we get:

y= 5x2 + 10x + 10 

⇒ y = 5(-1)2 + 10X(-1) + 10

⇒ y = 5 – 10 + 10

⇒ y = 5 

                       Therefore v(h,k)= v(-1,5). The answer matches for both methods.

To check how to convert from standard form to vertex form for the same equation, go to the link, and check the sample parabola solved in the process: How to convert Vertex form to Standard form.

Example (1): 

y = x2 + 4x – 2

The standard form:y = x2 + 4x – 2

So, in this equation, a=1, b=4, c=-2. (comparing with the standard equation y = ax2 + bx + c)

In this equation, the coefficient of x2 (or value of a) is already 1, therefore we won’t have to separate the number as a common factor.

Now, the coefficient of x (or the value of b) is 4. 

And, half of 4 is 2. 

So we add 22 on both sides of the equation.

 y + 22 = x2 + 4x – 2 + 22

⇒y + 22 = x2 + 4x + 22 – 2

⇒ y + 22  = (x+2)2 – 2 

⇒ y + 4 = (x+2)2 – 2

⇒ y = (x+2)2 – 2 -4

⇒y = (x+2)2 -6. This is the vertex form.

So the vertex is -2, -6.

a=1, h=-2, k=-6.  (comparing with the vertex equation y = a(x-h)2 + k).

We can verify the answer with the completing square method equation:

x = – b / 2a

⇒ x = –  4/ (2 X 1) (putting the value a=1, b=4)

⇒ x = –  2.

If we put the value x = -2 on the standard form equation, we get:

y = x2 + 4x – 2

⇒ y = (-2)2 + 4x(-2) -2

⇒ y = 4 -8 -2

⇒ y = 4-10

⇒ y = -6

Therefore v(h,k)= v(-2,-6). The answer matches for both methods.

To check how to convert from standard form to vertex form for the same equation, go to the link, and check Example (1): How to convert Vertex form to Standard form.

Example (2): 

y = -x2 + 6x + 5

The standard form: y = -x2 + 6x + 5

So, in this equation, a=-1, b=6, c=5. (comparing with the standard equation y = ax2 + bx + c)

In this equation, the coefficient of x2 (or value of a) is (-1), therefore we will have to separate (-1) as a common factor.

y = -x2 + 6x + 5

           ⇒y = – (x2 – 6x – 5)

Now, the coefficient of x (or the value of b) is (-6). 

And, half of (-6) is (-3)

So we add (-32) on both sides of the equation.

y + (-32) = – (x2 – 6x – 5) + (-32)

           ⇒ y – 32= – (x2 – 6x – 5) -32

           ⇒ y – 32= – (x2 – 6x – 5 + 32)

           ⇒ y – 32= – (x2 – 6x + 32 – 5)

           ⇒ y – 32= – {(x2 – 6x + 32) – 5}

           ⇒ y – 32= – {(x -3)2 – 5}

           ⇒ y – 9= – (x -3)2 + 5

           ⇒ y = – (x -3)2 + 5 + 9

           ⇒ y = – (x -3)2 + 14. This is the vertex form.

So the vertex is 3, 14.

a=-1, h=3, k=14.  (comparing with the vertex equation y = a(x-h)2 + k).

We can verify the answer with the completing square method equation: 

 x = – b / 2a

⇒ x = –  6/ {2 X (-1)}  (putting the value a=-1, b=6)

⇒ x = –  6/(-2)

⇒ x = 3

If we put the value x = 3 on the standard form equation, we get:

            y = -x2 + 6x + 5

            ⇒ y = – (3)2 + 6 X (3) + 5

            ⇒ y = – 9 + 18 + 5

            ⇒ y = 9 + 5

            ⇒ y = 14

Therefore v(h,k)= v(3,14). The answer matches for both methods.

To check how to convert from standard form to vertex form for the same equation, go to the link, and check Example (2): How to convert Vertex form to Standard form.