Question

the equation of a parabola is $y = x^2 + 2x + 1$. write the equation in vertex form.
write any numbers as integers or simplified proper or improper fractions.

Explanation:

Step1: Recall Vertex Form

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. We need to complete the square for the given quadratic \( y = x^2 + 2x + 1 \).

Step2: Complete the Square

Notice that \( x^2 + 2x + 1 \) is a perfect square trinomial. The formula for a perfect square trinomial is \( (x + m)^2 = x^2 + 2mx + m^2 \). Comparing with \( x^2 + 2x + 1 \), we see that \( 2m = 2 \) (so \( m = 1 \)) and \( m^2 = 1 \), which matches the constant term. So, \( x^2 + 2x + 1 = (x + 1)^2 \).

Thus, the equation \( y = x^2 + 2x + 1 \) in vertex form is \( y = (x + 1)^2 + 0 \) (or simply \( y = (x + 1)^2 \) since the \( + 0 \) is optional).

Answer:

\( y = (x + 1)^2 \)