General Form

f(x) = ax2 + bx + c

Standard Form

f(x) = a (x - h)2 + k

Transforming Quadratic Functions from General Form

to Standard Form

STEPS:1. Factor out a from the first two terms.

2. Complete the square.

3. Factor and combine.

f(x) = ax2 + bx + c

Step 1: Factor out a from the first two terms.

f(x) = a (x2 +𝑏𝑥

𝑎) + c

Step 2: Complete the square.

f(x) = a [x2 +bx + ( 𝑏

𝑎𝑥)2] + c - a(

𝑏

2𝑎)2

f(x) = a [x2 + bx +( 𝑏

𝑎𝑥)2] + c -

𝑎𝑏2

4𝑎2

f(x) = a [x2 + bx +( 𝑏

𝑎𝑥)2] + c -

𝑏2

4𝑎

Step 3: Factor and combine.

f(x) = a (x + 𝑏

2𝑎)2 +

4𝑎𝑐−𝑏2

4𝑎

h= -𝑏

2𝑎and k =

4𝑎𝑐−𝑏2

4𝑎

f(x) = a (x - h)2 + k

Examples

f(x) = x2 – 8x + 3 a = 1

f(x) = (x2 – 8x) + 3

f(x) = (x2 –8x + ( 8

2)2) + 3 - (

8

2)2

f(x) = (x2 – 8x + (4)2) + 3 – (4)2

f(x) = (x2 – 8x + 16) + 3 – 16

f(x) = (x – 4) 2 – 13

f(x) = 2 x2 + 5𝑥 – 1 a = 2

f(x) = 2 (x2 + 5

2𝑥 ) – 1

f(x) = 2 [x2 + 5

2𝑥 + (

25

16)] – 1 -

25

8

f(x) = 2 (x + 5

4)2 + (

−8−25

8)

f(x) = 2 (x + 5

4)2 + (

−33

8)

1. f(x) = x2 - 12x + 4

2. f(x) = 5x2 + 20x -12

3. f(x) = x2 + 3x + 36

4. f(x) = 4x2 + 2x + 9

5. f(x) = 3x2 + 6x + 2

ActivityTransform the following quadratic functions into standard form.