Question
Match each equation with the correct feature.
y = - 1/2 (x - 4)^2 + 1
y = (x + 4)^2 - 1
y = 3 (x - 4)^2 - 1
y = (x + 1)^2 + 4
Options:
Vertex at (4, -1)
Axis of symmetry at x = -4
Maximum at 1
Minimum at 4
y = - 1/2 (x - 4)^2 + 1
y = (x + 4)^2 - 1
y = 3 (x - 4)^2 - 1
y = (x + 1)^2 + 4
Options:
Vertex at (4, -1)
Axis of symmetry at x = -4
Maximum at 1
Minimum at 4
Answers
GPT-4o mini
Let's analyze each equation to match them with the corresponding features.
1. **Equation: \(y = -\frac{1}{2}(x - 4)^2 + 1\)**
- This equation is in vertex form \(y = a(x - h)^2 + k\) where \((h, k)\) is the vertex.
- Here, \(h = 4\) and \(k = 1\), so the vertex is at \((4, 1)\).
- The coefficient \(a = -\frac{1}{2}\) is negative, indicating that this parabola opens downwards, which means it has a maximum point.
- **Match: Vertex at \( (4, 1) \)** and **Maximum at 1**
2. **Equation: \(y = (x + 4)^2 - 1\)**
- This is also in vertex form, where \(h = -4\) and \(k = -1\). Thus, the vertex is at \((-4, -1)\).
- Since \(a = 1\) (positive), the parabola opens upwards, indicating a minimum point.
- **Match: Axis of symmetry at \(x = -4\)** and **Minimum at -1**
3. **Equation: \(y = 3(x - 4)^2 - 1\)**
- Again, in vertex form, with \(h = 4\) and \(k = -1\), the vertex is at \((4, -1)\).
- The coefficient \(a = 3\) is positive, meaning the parabola opens upwards and thus has a minimum point.
- **Match: Vertex at \( (4, -1) \)** and **Minimum at -1**
4. **Equation: \(y = (x + 1)^2 + 4\)**
- In vertex form, where \(h = -1\) and \(k = 4\), the vertex is at \((-1, 4)\).
- With \(a = 1\) (positive), this parabola opens upwards, giving it a minimum point.
- **Match: Minimum at 4**
**Summary of Matches:**
- **y = -1/2 (x - 4)^2 + 1**: Maximum at 1
- **y = (x + 4)^2 - 1**: Axis of symmetry at x = -4, Minimum at -1
- **y = 3 (x - 4)^2 - 1**: Vertex at (4, -1), Minimum at -1
- **y = (x + 1)^2 + 4**: Minimum at 4
Based on the problem characteristics, you may match the vertex and axis of symmetry accordingly.
Final corrected matches:
1. **\(y = -\frac{1}{2}(x - 4)^2 + 1\)** ----> **Maximum at 1**
2. **\(y = (x + 4)^2 - 1\)** ----> **Axis of symmetry at x = -4**
3. **\(y = 3(x - 4)^2 - 1\)** ----> **Vertex at (4, -1)**
4. **\(y = (x + 1)^2 + 4\)** ----> **Minimum at 4**
Thus matched correctly:
- Vertex at \((4, -1)\): for \(y = 3(x - 4)^2 - 1\)
- Axis of symmetry at \(x = -4\): for \(y = (x + 4)^2 - 1\)
- Maximum at \(1\): for \(y = -\frac{1}{2}(x - 4)^2 + 1\)
- Minimum at \(4\): for \(y = (x + 1)^2 + 4\)
1. **Equation: \(y = -\frac{1}{2}(x - 4)^2 + 1\)**
- This equation is in vertex form \(y = a(x - h)^2 + k\) where \((h, k)\) is the vertex.
- Here, \(h = 4\) and \(k = 1\), so the vertex is at \((4, 1)\).
- The coefficient \(a = -\frac{1}{2}\) is negative, indicating that this parabola opens downwards, which means it has a maximum point.
- **Match: Vertex at \( (4, 1) \)** and **Maximum at 1**
2. **Equation: \(y = (x + 4)^2 - 1\)**
- This is also in vertex form, where \(h = -4\) and \(k = -1\). Thus, the vertex is at \((-4, -1)\).
- Since \(a = 1\) (positive), the parabola opens upwards, indicating a minimum point.
- **Match: Axis of symmetry at \(x = -4\)** and **Minimum at -1**
3. **Equation: \(y = 3(x - 4)^2 - 1\)**
- Again, in vertex form, with \(h = 4\) and \(k = -1\), the vertex is at \((4, -1)\).
- The coefficient \(a = 3\) is positive, meaning the parabola opens upwards and thus has a minimum point.
- **Match: Vertex at \( (4, -1) \)** and **Minimum at -1**
4. **Equation: \(y = (x + 1)^2 + 4\)**
- In vertex form, where \(h = -1\) and \(k = 4\), the vertex is at \((-1, 4)\).
- With \(a = 1\) (positive), this parabola opens upwards, giving it a minimum point.
- **Match: Minimum at 4**
**Summary of Matches:**
- **y = -1/2 (x - 4)^2 + 1**: Maximum at 1
- **y = (x + 4)^2 - 1**: Axis of symmetry at x = -4, Minimum at -1
- **y = 3 (x - 4)^2 - 1**: Vertex at (4, -1), Minimum at -1
- **y = (x + 1)^2 + 4**: Minimum at 4
Based on the problem characteristics, you may match the vertex and axis of symmetry accordingly.
Final corrected matches:
1. **\(y = -\frac{1}{2}(x - 4)^2 + 1\)** ----> **Maximum at 1**
2. **\(y = (x + 4)^2 - 1\)** ----> **Axis of symmetry at x = -4**
3. **\(y = 3(x - 4)^2 - 1\)** ----> **Vertex at (4, -1)**
4. **\(y = (x + 1)^2 + 4\)** ----> **Minimum at 4**
Thus matched correctly:
- Vertex at \((4, -1)\): for \(y = 3(x - 4)^2 - 1\)
- Axis of symmetry at \(x = -4\): for \(y = (x + 4)^2 - 1\)
- Maximum at \(1\): for \(y = -\frac{1}{2}(x - 4)^2 + 1\)
- Minimum at \(4\): for \(y = (x + 1)^2 + 4\)