Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Match the function with its graph.**
Given Function:
\[ f(x) = -3(x+2)^2 \]
**Graph Analysis:**
The problem is to match the quadratic function \( f(x) = -3(x+2)^2 \) with one of the provided graphs. This function is a downward-opening parabola because the coefficient of the squared term is negative.
**Key Characteristics:**
- **Vertex**: The vertex form of a quadratic equation is \( f(x) = a(x-h)^2 + k \). For \( f(x) = -3(x+2)^2 \), the vertex is at (-2, 0).
- **Direction**: The parabola opens downwards due to the negative sign in front of the 3.
- **Stretch Factor**: The coefficient -3 indicates a vertical stretch compared to the standard parabola \( y = x^2 \).
**Description of Graphs:**
1. **Graph 1** (Top Left):
- Vertex: Appears to be at the origin (0,0).
- Opens downwards.
2. **Graph 2** (Top Center):
- Vertex: Close to (0, 0).
- Opens upwards.
3. **Graph 3** (Top Right):
- Vertex: Appears shifted left.
- Opens downwards.
4. **Graph 4** (Bottom Left):
- Vertex: Appears shifted right.
- Opens upwards.
**Conclusion:**
The correct graph for the function \( f(x) = -3(x+2)^2 \) is **Graph 3** (Top Right). This graph has a vertex shifted left from the origin, aligns with the downward opening, and fits the description of a vertical stretch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb10ec9c2-0bba-4463-a834-dec14478872b%2F0ba7d670-176f-47fe-beda-e57e66adfb82%2Fvbf9nt8_processed.png&w=3840&q=75)
Transcribed Image Text:**Match the function with its graph.**
Given Function:
\[ f(x) = -3(x+2)^2 \]
**Graph Analysis:**
The problem is to match the quadratic function \( f(x) = -3(x+2)^2 \) with one of the provided graphs. This function is a downward-opening parabola because the coefficient of the squared term is negative.
**Key Characteristics:**
- **Vertex**: The vertex form of a quadratic equation is \( f(x) = a(x-h)^2 + k \). For \( f(x) = -3(x+2)^2 \), the vertex is at (-2, 0).
- **Direction**: The parabola opens downwards due to the negative sign in front of the 3.
- **Stretch Factor**: The coefficient -3 indicates a vertical stretch compared to the standard parabola \( y = x^2 \).
**Description of Graphs:**
1. **Graph 1** (Top Left):
- Vertex: Appears to be at the origin (0,0).
- Opens downwards.
2. **Graph 2** (Top Center):
- Vertex: Close to (0, 0).
- Opens upwards.
3. **Graph 3** (Top Right):
- Vertex: Appears shifted left.
- Opens downwards.
4. **Graph 4** (Bottom Left):
- Vertex: Appears shifted right.
- Opens upwards.
**Conclusion:**
The correct graph for the function \( f(x) = -3(x+2)^2 \) is **Graph 3** (Top Right). This graph has a vertex shifted left from the origin, aligns with the downward opening, and fits the description of a vertical stretch.
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