Quadratic functions & equations

there are three types of forms.

Forms

Standard(general) form

f(x) = ax^2 + bx + c

e.g x^2 + 2x + 4

vertex form

y=a(x-h)^2+k

e.g (x+2)^2

factored form

(x+2)(x+2)

Vertex formula

https://www.tes.com/lessons/txuiCSZK6FGxcg/quadratic-formula-vertex-formula

h = -(b/2a)

k = -(b^2/4a) + c

proof

y = ax^2 + bx + c

= a(x^2 + bx/a) + c

= a{ (x + b/2a)^2 - b^2/4a^2 } + c

= a(x + b/2a)^2 - b^2/4a + c

Vertext form

y=a(x-h)^2+k

x-vertex = -b/2a

y-vertex = -b^2/4a + c

example

f(x)=8x^2 + 16x + 3

= 8(x^2 + 16x/8) + 3

= 8{ (x + 16/16)^2 - (16/16)^2 } + 3

= 8{ (x + 1)^2 - 1^2 } + 3

= 8{ (x + 1)^2 - 1^2 } + 3

= 8{ (x + 1)^2 } - 8(1^2) + 3

= 8(x + 1)^2 - 8(1^2) + 3

= 8(x + 1)^2 - 8 + 3

= 8(x + 1)^2 - 5

A parabola is defined as 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 for 𝑎 ≠ 0

By factoring 𝑎 and completing the square, we get

𝑦 = 𝑎(𝑥² + (𝑏 ∕ 𝑎)𝑥) + 𝑐 =

= 𝑎(𝑥 + 𝑏 ∕ (2𝑎))² + 𝑐 − 𝑏² ∕ (4𝑎)

With ℎ = −𝑏 ∕ (2𝑎) and 𝑘 = 𝑐 − 𝑏² ∕ (4𝑎) we get 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘

(𝑥 − ℎ)² ≥ 0 for all 𝑥 So the parabola will have a vertex when (𝑥 − ℎ)² = 0 ⇔ 𝑥 = ℎ ⇒ 𝑦 = 𝑘

𝑎 > 0 ⇒ (ℎ, 𝑘) is the minimum point. 𝑎 < 0 ⇒ (ℎ, 𝑘) is the maximum point.

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:standard-form-quadratic/v/ex3-completing-the-square

quadratic formula

Quadratic formula - Alchetron, The Free Social EncyclopediaAlchetron.com