Question

modeling and analyzing quadratic functions: quadratic functions
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^2 + 2x)^2 + 3 )
( f(x) = (x + 1)^2 + 2 )
( f(x) = (x + 3)^2 + 2x )

Explanation:

Step1: Identify the quadratic function

The given function is $f(x) = x^2 + 2x + 3$.

Step2: Complete the square for $x^2+2x$

Take the coefficient of $x$, which is 2. Half of it is 1, and squaring it gives $1^2=1$. Rewrite the function:
$$f(x) = (x^2 + 2x + 1) - 1 + 3$$

Step3: Simplify the expression

The first three terms form a perfect square, and combine constants:
$$f(x) = (x+1)^2 + 2$$

Answer:

B. $f(x) = (x + 1)^2 + 2$