Question
The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1.
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 (2, 10), has a vertex at (5, 1), and goes through (8, 10).
On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 8), has a vertex at (negative 5, negative 11), and goes through (negative 2, 8).
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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 (2, 10), has a vertex at (5, 1), and goes through (8, 10).
On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 8), has a vertex at (negative 5, negative 11), and goes through (negative 2, 8).
Mark this and return
Answers
GPT-4o mini
To determine which graph represents \( g(x) = (x - 5)^2 + 1 \), we need to understand the transformations applied to the original function \( f(x) = x^2 \).
1. **Vertex Translation**:
- The original parabola \( f(x) = x^2 \) has a vertex at (0, 0).
- The transformation \( g(x) = (x - 5)^2 + 1 \) means the parabola is shifted to the right by 5 units and up by 1 unit.
- Consequently, the vertex of \( g(x) \) is at \( (5, 1) \).
2. **Direction of Opening**:
- Since the leading coefficient of \( (x - 5)^2 \) is positive, the parabola opens upward.
3. **Points on the Graph**:
- To find additional points on the graph of \( g(x) \), we can calculate:
- When \( x = 5 \), \( g(5) = (5 - 5)^2 + 1 = 1 \) (the vertex).
- When \( x = 4 \), \( g(4) = (4 - 5)^2 + 1 = 2 \).
- When \( x = 6 \), \( g(6) = (6 - 5)^2 + 1 = 2 \).
- When \( x = 3 \), \( g(3) = (3 - 5)^2 + 1 = 5 \).
- When \( x = 7 \), \( g(7) = (7 - 5)^2 + 1 = 5 \).
Based on this information, we now check the options provided:
1. **First Option**:
- Vertex at \( (5, 1) \), goes through \( (2, 10) \) and \( (8, 10) \). **This option seems to match** as it has the vertex at (5, 1).
2. **Second Option**:
- Vertex at \( (5, -11) \). **Does not match**.
3. **Third Option**:
- Vertex at \( (-5, 1) \). **Does not match**.
4. **Fourth Option**:
- Vertex at \( (-5, -11) \). **Does not match**.
**Conclusion**: The graph representing \( g(x) \) is the **first option**, which has a vertex at \( (5, 1) \) and opens upward, consistent with the transformations applied to \( f(x) \).
1. **Vertex Translation**:
- The original parabola \( f(x) = x^2 \) has a vertex at (0, 0).
- The transformation \( g(x) = (x - 5)^2 + 1 \) means the parabola is shifted to the right by 5 units and up by 1 unit.
- Consequently, the vertex of \( g(x) \) is at \( (5, 1) \).
2. **Direction of Opening**:
- Since the leading coefficient of \( (x - 5)^2 \) is positive, the parabola opens upward.
3. **Points on the Graph**:
- To find additional points on the graph of \( g(x) \), we can calculate:
- When \( x = 5 \), \( g(5) = (5 - 5)^2 + 1 = 1 \) (the vertex).
- When \( x = 4 \), \( g(4) = (4 - 5)^2 + 1 = 2 \).
- When \( x = 6 \), \( g(6) = (6 - 5)^2 + 1 = 2 \).
- When \( x = 3 \), \( g(3) = (3 - 5)^2 + 1 = 5 \).
- When \( x = 7 \), \( g(7) = (7 - 5)^2 + 1 = 5 \).
Based on this information, we now check the options provided:
1. **First Option**:
- Vertex at \( (5, 1) \), goes through \( (2, 10) \) and \( (8, 10) \). **This option seems to match** as it has the vertex at (5, 1).
2. **Second Option**:
- Vertex at \( (5, -11) \). **Does not match**.
3. **Third Option**:
- Vertex at \( (-5, 1) \). **Does not match**.
4. **Fourth Option**:
- Vertex at \( (-5, -11) \). **Does not match**.
**Conclusion**: The graph representing \( g(x) \) is the **first option**, which has a vertex at \( (5, 1) \) and opens upward, consistent with the transformations applied to \( f(x) \).