Answer:
Explanation:
Given:
[tex]g(x)=3x^2+12x+9[/tex]
B.)
To find the vertex, we use the following formula:
[tex]\begin{gathered} x=-\frac{b}{2a} \\ \text{From the Form:} \\ y=ax^2+bx+c \end{gathered}[/tex]
So based on the given equation, the values of a and b are:
a=3
b=12
We plug in what we know:
[tex]\begin{gathered} x=-\frac{b}{2a} \\ =-\frac{12}{2(3)} \\ x=-2 \end{gathered}[/tex]
Next, we plug in x=-2 into g(x)=3x^2+12x+9:
[tex]\begin{gathered} g(x)=3x^2+12x+9 \\ =3(-2)^2+12(-2)+9 \\ \text{Calculate} \\ g(x)=-3 \end{gathered}[/tex]
Therefore, the vertex is (-2,-3).
C.
Now, to find the axis of symmetry, we also use the formula x=-b/2a since it is the vertical line that goes through the vertex.
Therefore, the Axis of Symmetry for the given equation is x = -2.
D.
We let g(x)=0 to find the x-intercept:
[tex]\begin{gathered} 3x^2+12x+9=0 \\ \text{Simplify} \\ =3(x^2+4x+3) \\ =3(x+1)(x+3) \end{gathered}[/tex]
Based on the factors, the values for x are:
x=-1
x=-3
Therefore, the x intercept points are:
(-1,0),(-3,0)
To get the y-intercept, we let x=0 and plug in into g(x)=3x^2+12x+9. So,
[tex]\begin{gathered} g(x)=3x^2+12x+9 \\ g(0)=3(0)^2+12(0)+9 \\ g(0)=9 \end{gathered}[/tex]
Therefore, the y intercept point is (0,9).
