Determine the vertex form of g(x) = x2 + 2x – 1. Which graph represents g(x)?

Question

On a coordinate plane, a parabola opens up. It goes through (negative 2, 3), has a vertex at (negative 1, 2), and goes through (0, 3).

On a coordinate plane, a parabola opens up. It goes through (negative 1, 2), has a vertex at (1, negative 2), and goes through (3, 2).

On a coordinate plane, a parabola opens up. It goes through (0, 3), has a vertex at (1, 2), and goes through (2, 3).

On a coordinate plane, a parabola opens up. It goes through (negative 3, 2), has a vertex at (negative 1, negative 2), and goes through (1, 2).

Answer 1

To convert the quadratic function \( g(x) = x^2 + 2x - 1 \) into vertex form, we can complete the square.

1. Start with the original equation: \[ g(x) = x^2 + 2x - 1 \]

2. Focus on the quadratic and linear terms \( x^2 + 2x \). To complete the square, take half of the coefficient of \( x \) (which is 2), square it, and add and subtract this value inside the equation: \[ \text{Half of 2 is 1, and } 1^2 = 1. \] Rewrite \( g(x) \): \[ g(x) = (x^2 + 2x + 1) - 1 - 1 \] \[ g(x) = (x + 1)^2 - 2 \]

Now we have the function in vertex form: \[ g(x) = (x + 1)^2 - 2 \] This shows that the vertex of the parabola is at the point \( (-1, -2) \).

To summarize:

• The vertex is at \( (-1, -2) \).

Now let's compare this vertex with the options provided for their corresponding vertices:

• Option 1: Vertex at \( (-1, 2) \) - Incorrect
• Option 2: Vertex at \( (1, -2) \) - Incorrect
• Option 3: Vertex at \( (1, 2) \) - Incorrect
• Option 4: Vertex at \( (-1, -2) \) - This is a match!

Thus, the correct graph that represents \( g(x) = x^2 + 2x - 1 \) is the one with the vertex at \( (-1, -2) \).