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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![### Completing the Square and Writing in Graphing Form
**5. Complete the square and write into graphing form:**
#### a. \( y = x^2 - 2x + 2 \)
**Vertex:**
**Stretch:**
**Opens:**
#### b. \( y = 2x^2 - 4x \)
**Vertex:**
**Stretch:**
**Opens:**
### Instructions:
1. To convert each quadratic equation into graphing (vertex) form, complete the square.
2. Identify and write the vertex coordinates.
3. Identify how the parabola stretches.
4. Determine the direction in which the parabola opens.
### Example:
For \( y = x^2 - 2x + 2 \):
1. **Complete the square:**
\[ y = (x^2 - 2x + 1) + 2 - 1 \]
\[ y = (x-1)^2 + 1 \]
2. **Graphing form:**
\[ y = (x-1)^2 + 1 \]
3. **Identify the vertex:**
\[ Vertex: (1, 1) \]
4. **Determine the parabola's stretch and direction:**
- **Stretch:** Standard parabola stretch (no coefficient change).
- **Opens:** Upward (positive coefficient of \(x^2\)).
For \( y = 2x^2 - 4x \):
1. **Complete the square:**
\[ y = 2(x^2 - 2x) \]
Add and subtract 1 inside the parenthesis:
\[ y = 2(x^2 - 2x + 1 - 1) \]
\[ y = 2((x-1)^2 - 1) \]
\[ y = 2(x-1)^2 - 2 \]
2. **Graphing form:**
\[ y = 2(x-1)^2 - 2 \]
3. **Identify the vertex:**
\[ Vertex: (1, -2) \]
4. **Determine the parabola's stretch and direction:**
- **Stretch:** Vertically stretched by a factor of 2.
- **Opens:** Upward (positive coefficient of \(2x^2\)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0e3b137-93c3-4f8a-ba86-2ec6c73db472%2Ff2d8e142-03dc-4a20-a785-1c1f5df21aef%2Fi0jbvs_processed.png&w=3840&q=75)
Transcribed Image Text:### Completing the Square and Writing in Graphing Form
**5. Complete the square and write into graphing form:**
#### a. \( y = x^2 - 2x + 2 \)
**Vertex:**
**Stretch:**
**Opens:**
#### b. \( y = 2x^2 - 4x \)
**Vertex:**
**Stretch:**
**Opens:**
### Instructions:
1. To convert each quadratic equation into graphing (vertex) form, complete the square.
2. Identify and write the vertex coordinates.
3. Identify how the parabola stretches.
4. Determine the direction in which the parabola opens.
### Example:
For \( y = x^2 - 2x + 2 \):
1. **Complete the square:**
\[ y = (x^2 - 2x + 1) + 2 - 1 \]
\[ y = (x-1)^2 + 1 \]
2. **Graphing form:**
\[ y = (x-1)^2 + 1 \]
3. **Identify the vertex:**
\[ Vertex: (1, 1) \]
4. **Determine the parabola's stretch and direction:**
- **Stretch:** Standard parabola stretch (no coefficient change).
- **Opens:** Upward (positive coefficient of \(x^2\)).
For \( y = 2x^2 - 4x \):
1. **Complete the square:**
\[ y = 2(x^2 - 2x) \]
Add and subtract 1 inside the parenthesis:
\[ y = 2(x^2 - 2x + 1 - 1) \]
\[ y = 2((x-1)^2 - 1) \]
\[ y = 2(x-1)^2 - 2 \]
2. **Graphing form:**
\[ y = 2(x-1)^2 - 2 \]
3. **Identify the vertex:**
\[ Vertex: (1, -2) \]
4. **Determine the parabola's stretch and direction:**
- **Stretch:** Vertically stretched by a factor of 2.
- **Opens:** Upward (positive coefficient of \(2x^2\)).
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