Answer
graph of $f(x)=x^2+4x+3$
Work Step by Step
In the form $f(x)=a(x-h)^2+k,$ the given equation, $f(x)=x^2+4x+3,$ is equivalent to
\begin{align*}
f(x)&=(x^2+4x)+3
\\\\&=
\left(x^2+4x+\left(\dfrac{4}{2}\right)^2 \right)+3-\left(\dfrac{4}{2}\right)^2
&(\text{complete the square})
\\\\&=
\left(x^2+4x+4 \right)+3-4
\\&=
(x+2)^2-1
\\&=
[x-(-2)]^2-1
.\end{align*}
The vertex of the given quadratic function is $(-2,-1).$
Substituting $x=-4,-3,-1,0,$ then the corresponding values of $y$ are
\begin{align*}
\text{If }x=-4:f(x)&=[x-(-2)]^2-1
\\f(-4)&=[-4-(-2)]^2-1
\\&=
[-4+2]^2-1
\\&=
[-2]^2-1
\\&=
4-1
\\&=
3
\\\\
\text{If }x=-3:f(x)&=[x-(-2)]^2-1
\\f(-3)&=[-3-(-2)]^2-1
\\&=
[-3+2]^2-1
\\&=
[-1]^2-1
\\&=
1-1
\\&=
0
\\\\
\text{If }x=-1:f(x)&=[x-(-2)]^2-1
\\f(-1)&=[-1-(-2)]^2-1
\\&=
[-1+2]^2-1
\\&=
[1]^2-1
\\&=
1-1
\\&=
0
\\\\
\text{If }x=0:f(x)&=[x-(-2)]^2-1
\\f(0)&=[0-(-2)]^2-1
\\&=
[0+2]^2-1
\\&=
4-1
\\&=
3
.\end{align*}
Hence, the graph of $f(x)=x^2+4x+3$ is a parabola with the vertex at $(-2,-1)$. It passes through $(-4,3), (-3,0), (-1,0),$ and $(0,3)$.
