The image shows a geometric representation of the function f(x) = x^2 + 2x + 3 written in standard form. what is this function written in vertex form?

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

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

f(x) = (x + 1)^2 + 2

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

f(x) = (x + 1)² + 2 in vertex form ⇒ 3rd answer

Step-by-step explanation:

* Lets revise how to find the vertex form the standard form

- Standard form ⇒ x² + bx + c, where a , b , c are constant

- Vertex form ⇒(x - h)² + k, where h , k are constant and  (h , k) is the

  vertex point (minimum or maximum)  of the function

- At first we must find h and k

- By equating the two forms we can find the value of h and k

* Lets solve the problem

∵ f(x) = x² + 2x + 3 ⇒ standard form

∵ f(x) = (x - h)² + k ⇒ vertex form

- Put them equal each other

∴ x² + 2x + 3 = (x - h)² + k ⇒ open the bracket power 2

∴ x² + 2x + 3 = x² - 2hx + h² + k

- Now compare the like terms in both sides

∵ 2x = -2hx ⇒ cancel x from both sides

∴ 2 = -2h ⇒ divide both sides by -2

∴ -1 = h

∴ The value of h is -1

∵ 3 = h² + k

- Substitute the value of h

∴ 3 = (-1)² + k

∴ 3 = 1 + k ⇒ subtract 1 from both sides

∴ 2 = k

∴ The value of k = 2

- Lets substitute the value of h and k in the vertex form

∴ f(x) = (x - -1)² + 2

∴ f(x) = (x + 1)² + 2

* f(x) = (x + 1)² + 2 in vertex form

think about it deeply and passionately about this property

i am sorry