This document describes the process of transforming a quadratic function from general form to standard form. It shows the general forms of quadratic functions, and the three step process to transform them: 1) factor out the leading coefficient a, 2) complete the square, 3) factor and combine terms. It provides examples of applying these steps to functions like f(x) = x^2 - 8x + 3 and f(x) = 2x^2 + 5x - 1. Finally, it lists 5 additional quadratic functions to transform into standard form.

2. General Form
f(x) = ax2 + bx + c

3. Standard Form
f(x) = a (x - h)2 + k

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
f(x) = a [x2 + bx +(
𝑏
𝑎
𝑥)2] + c -
𝑏2
4𝑎

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.