Final answer:
The standard form of the quadratic function y = 3(x+1)^2 + 4 is y = 3x^2 + 6x + 7, which is accomplished by expanding the binomial and distributing the coefficient 3, followed by adding 4.
Step-by-step explanation:
To write the given quadratic function in standard form, we need to expand the binomial (x+1)2 and then distribute the coefficient 3 across the expanded terms. The equation y = 3(x+1)2 + 4 is already given to us in vertex form. Let's expand it step by step.
Firstly, the binomial (x+1)2 needs to be expanded using the formula (a+b)2 = a2 + 2ab + b2:
(x+1)2 = x2 + 2(x)(1) + 12
(x+1)2 = x2 + 2x + 1
Now we apply the coefficient 3 to each term in the expanded binomial:
3(x2 + 2x + 1) = 3x2 + 6x + 3
Finally, we add 4 to complete the transformation:
y = 3x2 + 6x + 3 + 4
y = 3x2 + 6x + 7
Therefore, the standard form of the given quadratic function is y = 3x2 + 6x + 7, which corresponds to Option 1.