This document describes how to transform quadratic functions from general form to standard form in three steps: 1) Factor out the leading coefficient a from the first two terms. 2) Complete the square of the factored quadratic expression. 3) Factor the completed square and combine like terms. Examples of transforming specific quadratic functions like f(x) = x^2 - 8x + 3 and f(x) = 2x^2 + 5x - 1 are provided. An activity with 5 quadratic functions to transform to standard form is listed at the end.

2. General Form
f(x) = ax2
c

3. Standard
Form
f(x) = a (x

4. Transforming
Quadratic Functions
from
General Form
to Standard Form

5. STEPS:
1. Factor out a from the first
two terms.
2. Complete the square.
3. Factor and combine.

6. f(x) = ax2 + bx + c
Step 1: Factor out a from the first
two terms.
f(x) = a (x2 +
𝑏𝑥
𝑎
) + c

7. Step 2: Complete the square.
f(x) = a [x2 +bx + (
𝑏
𝑎
𝑥)2] + c - a(
𝑏
2𝑎
)2
f(x) = a [x2 + bx +(
𝑏
𝑎
𝑥)2] + c -
𝑎𝑏2
4𝑎2
2 𝑏 2 𝑏2

8. Step 3: Factor and combine.
f(x) = a (x +
𝑏
2𝑎
)2 +
4𝑎𝑐−𝑏2
4𝑎
h= -
𝑏
2𝑎
and k =
4𝑎𝑐−𝑏2
4𝑎
f(x)=a(x-h)2 + k

9. Examples

10. 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

11. f(x) = 2 x2 + 5𝑥 – 1 a = 2
f(x) = 2 (x2 +
5
2
𝑥 ) – 1
f(x) = 2 [x2 +
5
2
𝑥 + (
25
16
)] – 1 -
25
8
f(x) = 2 (x+
5
4
)2 + (
−8−25
8
)
f(x) = 2 (x+
5
4
)2 + (
−33
8
)

12. 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
Activity
Transform the following quadratic functions
into standard form.