Convert the Function f(x)=(x-2)²+3 to its Standard Form

To convert the function from vertex form to standard form, follow these steps:

• Step 1: Identify the vertex form - $f(x) = (x-2)^2 + 3$. The terms inside the parentheses represent a perfect square trinomial.
• Step 2: Expand the square. Recall: $(a-b)^2 = a^2 - 2ab + b^2$. Here, $a = x$ and $b = 2$.
• Step 3: Expand $(x-2)^2$:
  $(x-2)^2 = x^2 - 2 \cdot x \cdot 2 + 2^2 = x^2 - 4x + 4$.
• Step 4: Add the constant from the original function:
  $f(x) = (x^2 - 4x + 4) + 3 = x^2 - 4x + 7$.

After expanding and simplifying, we find that $f(x) = x^2 - 4x + 7$ is the standard form of the function.

Therefore, the correct choice that matches this solution is choice 3, which is $f(x) = x^2 - 4x + 7$.