Question
Determine the vertex form of g(x) = x2 + 2x – 1. Which graph represents g(x)?
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).
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).
Answers
GPT-4o mini
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 \( (-1, -2) \).
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 \( (-1, -2) \).