Question

determine the vertex form of ( g(x) = x^2 + 2x - 1 ). which graph represents ( g(x) )?

Explanation:

Step1: Complete the square for $g(x)$

Start with $g(x) = x^2 + 2x - 1$. Group the $x$-terms:
$g(x) = (x^2 + 2x) - 1$
Add and subtract $(\frac{2}{2})^2=1$ inside the parentheses:
$g(x) = (x^2 + 2x + 1 - 1) - 1$

Step2: Rewrite in vertex form

Factor the perfect square trinomial and simplify constants:
$g(x) = (x+1)^2 - 1 - 1$
$g(x) = (x+1)^2 - 2$

Step3: Identify vertex and direction

The vertex form $g(x)=a(x-h)^2+k$ has vertex $(h,k)=(-1,-2)$. Since $a=1>0$, the parabola opens upward.

Step4: Match to the graph

The second graph has a vertex at $(-1,-2)$ and opens upward, matching the function.

Answer:

Vertex form: $g(x) = (x+1)^2 - 2$
The correct graph is the middle (second) one.