Convert the rectangular equation to polar form and sketch its graph. y² = 4x -4 -4 -2 -2 π/2 4 2 -2 π/2 4 2 -2 2 2 4 4 0 0 -4 -4 -2 -2 π/2 4 2 -2 π/2 4 2 2 2 4 4 0 0
Convert the rectangular equation to polar form and sketch its graph. y² = 4x -4 -4 -2 -2 π/2 4 2 -2 π/2 4 2 -2 2 2 4 4 0 0 -4 -4 -2 -2 π/2 4 2 -2 π/2 4 2 2 2 4 4 0 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![# Converting Rectangular Equation to Polar Form
## Problem Statement:
Convert the rectangular equation to polar form and sketch its graph.
## Equation Given:
\[ y^2 = 4x \]
### Graphs of Possible Polar Representations:
#### Graph Descriptions:
1. **Graph (Top Left)**:
- This graph depicts a curve opening to the right. The curve passes through the point \( (0,0) \) and extends to positive x-values.
2. **Graph (Top Right)**:
- This graph shows a parabolic curve opening upwards. The vertex of the parabola is at the origin \( (0,0) \), and it extends in both positive and negative directions along the x-axis.
3. **Graph (Bottom Left)**:
- This graph presents a curve opening to the left. The curve passes through the origin \( (0,0) \) and expands to negative x-values.
4. **Graph (Bottom Right)**:
- This graph illustrates another parabolic curve opening upwards, similar to the top right graph. The curve passes through the origin \( (0,0) \).
### Explanation of Correct Graph:
To identify which graph corresponds to the equation \( y^2 = 4x \) after converting to polar coordinates:
1. The standard form for a parabola opening to the right is \( y^2 = 4ax \), indicating that it would exhibit symmetry along the y-axis and open towards the positive x-direction.
2. The transformation into polar coordinates does not break this symmetry.
Therefore, the correct graph for the given equation is the **Top Left Graph**.
### Answer Key:
- The blue circle (o) next to the top left graph indicates this graph is the correct portrayal for the given equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F894ba220-0d5e-4e50-906d-a5e96c05bfb6%2F08ab6c29-d4e4-4116-92c9-93808f410cdc%2Fykyt2hj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Converting Rectangular Equation to Polar Form
## Problem Statement:
Convert the rectangular equation to polar form and sketch its graph.
## Equation Given:
\[ y^2 = 4x \]
### Graphs of Possible Polar Representations:
#### Graph Descriptions:
1. **Graph (Top Left)**:
- This graph depicts a curve opening to the right. The curve passes through the point \( (0,0) \) and extends to positive x-values.
2. **Graph (Top Right)**:
- This graph shows a parabolic curve opening upwards. The vertex of the parabola is at the origin \( (0,0) \), and it extends in both positive and negative directions along the x-axis.
3. **Graph (Bottom Left)**:
- This graph presents a curve opening to the left. The curve passes through the origin \( (0,0) \) and expands to negative x-values.
4. **Graph (Bottom Right)**:
- This graph illustrates another parabolic curve opening upwards, similar to the top right graph. The curve passes through the origin \( (0,0) \).
### Explanation of Correct Graph:
To identify which graph corresponds to the equation \( y^2 = 4x \) after converting to polar coordinates:
1. The standard form for a parabola opening to the right is \( y^2 = 4ax \), indicating that it would exhibit symmetry along the y-axis and open towards the positive x-direction.
2. The transformation into polar coordinates does not break this symmetry.
Therefore, the correct graph for the given equation is the **Top Left Graph**.
### Answer Key:
- The blue circle (o) next to the top left graph indicates this graph is the correct portrayal for the given equation.
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