Question
The graph of f(x) = x2 is translated to form
g(x) = (x – 2)2 – 3.
On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4).
Which graph represents g(x)?
On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1).
On a coordinate plane, a parabola opens up. It goes through (negative 3, 4), has a vertex at (negative 2, 3), and goes through (negative 1, 4).
On a coordinate plane, a parabola opens up. It goes through (1, 4), has a vertex at (2, 3), and goes through (3, 4).
On a coordinate plane, a parabola opens up. It goes through (negative 4, 1), has a vertex at (negative 2, negative 3), and goes through (0, 1).
g(x) = (x – 2)2 – 3.
On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4).
Which graph represents g(x)?
On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1).
On a coordinate plane, a parabola opens up. It goes through (negative 3, 4), has a vertex at (negative 2, 3), and goes through (negative 1, 4).
On a coordinate plane, a parabola opens up. It goes through (1, 4), has a vertex at (2, 3), and goes through (3, 4).
On a coordinate plane, a parabola opens up. It goes through (negative 4, 1), has a vertex at (negative 2, negative 3), and goes through (0, 1).
Answers
GPT-4o mini
To determine which graph represents \( g(x) = (x - 2)^2 - 3 \), let's analyze the transformation applied to \( f(x) = x^2 \).
1. **Find the vertex**:
The given function \( g(x) = (x - 2)^2 - 3 \) is in vertex form, \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( g(x) \):
- \( h = 2 \)
- \( k = -3 \)
Therefore, the vertex of \( g(x) \) is at the point \((2, -3)\).
2. **Find the direction**:
The coefficient in front of the squared term is positive (1), meaning that the parabola opens upwards.
3. **Identify points**:
To find other points on the graph, we can substitute some values into \( g(x) \):
- When \( x = 2 \):
\[
g(2) = (2 - 2)^2 - 3 = 0 - 3 = -3 \quad (\text{Vertex at } (2, -3))
\]
- When \( x = 1 \):
\[
g(1) = (1 - 2)^2 - 3 = 1 - 3 = -2
\]
- When \( x = 3 \):
\[
g(3) = (3 - 2)^2 - 3 = 1 - 3 = -2
\]
- When \( x = 0 \):
\[
g(0) = (0 - 2)^2 - 3 = 4 - 3 = 1
\]
- When \( x = 4 \):
\[
g(4) = (4 - 2)^2 - 3 = 4 - 3 = 1
\]
This tells us that the points \((0, 1)\) and \((4, 1)\) both lie on the graph of \( g(x) \).
4. **Match with given options**:
Now let's match \( g(x) \) with the provided graph options:
- The first option has a vertex at \((2, -3)\) and points \((0, 1)\) and \((4, 1)\).
Therefore, the correct graph representing \( g(x) = (x - 2)^2 - 3 \) is the first option titled "On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1)."
1. **Find the vertex**:
The given function \( g(x) = (x - 2)^2 - 3 \) is in vertex form, \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( g(x) \):
- \( h = 2 \)
- \( k = -3 \)
Therefore, the vertex of \( g(x) \) is at the point \((2, -3)\).
2. **Find the direction**:
The coefficient in front of the squared term is positive (1), meaning that the parabola opens upwards.
3. **Identify points**:
To find other points on the graph, we can substitute some values into \( g(x) \):
- When \( x = 2 \):
\[
g(2) = (2 - 2)^2 - 3 = 0 - 3 = -3 \quad (\text{Vertex at } (2, -3))
\]
- When \( x = 1 \):
\[
g(1) = (1 - 2)^2 - 3 = 1 - 3 = -2
\]
- When \( x = 3 \):
\[
g(3) = (3 - 2)^2 - 3 = 1 - 3 = -2
\]
- When \( x = 0 \):
\[
g(0) = (0 - 2)^2 - 3 = 4 - 3 = 1
\]
- When \( x = 4 \):
\[
g(4) = (4 - 2)^2 - 3 = 4 - 3 = 1
\]
This tells us that the points \((0, 1)\) and \((4, 1)\) both lie on the graph of \( g(x) \).
4. **Match with given options**:
Now let's match \( g(x) \) with the provided graph options:
- The first option has a vertex at \((2, -3)\) and points \((0, 1)\) and \((4, 1)\).
Therefore, the correct graph representing \( g(x) = (x - 2)^2 - 3 \) is the first option titled "On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1)."