Question

which is the graph of the function $f(x)=x^2 + 2x + 3$?

Explanation:

Step1: Find vertex of parabola

Rewrite function in vertex form:

$$\begin{align*} f(x)&=x^2+2x+3\\ &=(x^2+2x+1)+2\\ &=(x+1)^2+2 \end{align*}$$

Vertex is $(-1, 2)$.

Step2: Identify y-intercept

Set $x=0$:
$$f(0)=0^2+2(0)+3=3$$
Y-intercept is $(0, 3)$.

Step3: Match to graph

Check which graph has vertex $(-1,2)$ and passes through $(0,3)$. The top graph matches these points.

Answer:

The top graph (the first grid with the parabola opening upward, vertex at $(-1, 2)$ and passing through $(0, 3)$)