y = x² + 6x + 3
y = x² + 6x + 9 - 6
y = (x + 3)² - 6
since the coefficient of x² is positive, then min at (-3,-6). the answer is C.
Answer:
Step-by-step explanation:
The given expression is:
[tex]y=x^{2}+6x+3[/tex]
So, to complete the square, we first have to find the squared quotient between the second-term coefficient and 2, the add and subtract this number at the same time in the expression, like this:
[tex]b=6[/tex]
[tex](\frac{b}{2} )^{2}=(\frac{6}{2} )^{2}=9[/tex]
Then,
[tex]y=x^{2}+6x+3+9-9[/tex]
Now, we just have to group the terms that can be factorized:
[tex]y=(x^{2}+6x+9)+3-9[/tex]
Then, the factorization would be made using the squared root of [tex]x^{2}[/tex] and [tex]9[/tex], which is:
[tex]y=(x+3)^{2} +3-9[/tex]
At the end, we just operate independent number outside the factorization:
[tex]y=(x+3)^{2} -6[/tex]
So, according to the complete square of the expression, we can see that the vertex has coordinates [tex](-3;-6)[/tex], which is a minimum, because the squared coefficient is positive, that means the parabola is concave up.
The reason why the vertex has that coordinates is because the explicit expression of a parabola is:
[tex]y=(x-h)^{2} +k[/tex]
Where [tex](h;k)[/tex] is vertex coordinates.
Therefore, the answer is C.
