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Transforming Quadratic functions from General Form f(x)=ax2+bx+c to Standard Form f(x)=a(x-h)2+k
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General Form
f(x) = ax2 + bx
c
Standard Form
f(x) = a (x -
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 +( 𝑏𝑥)2] + c -
𝑏2
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 –16f(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 + 42. f(x) = 5x2 + 20x -123. f(x) = x2 + 3x + 364. f(x) = 4x2 + 2x + 95. f(x) = 3x2 + 6x + 2
ActivityTransform the following quadratic functions
into standard form.