Question
Consider the Cartesian equation of a circle, y equals negative 2 plus or minus Start Root 36 minus left parenthesis x minus 1 right parenthesis squared End Root, versus the parametric equations of the same circle:
x = 1 + 6 cos t
y equals negative 2 plus 6 sine t.
What are the advantages of the parametric equations? Select all that apply.
(2 points)
Responses
The parametric equations show the direction in which the circle is drawn.
The parametric equations show the direction in which the circle is drawn.
The parametric equations describe a three-dimensional figure.
The parametric equations describe a three-dimensional figure.
For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
The parametric equations have a finite domain.
The parametric equations have a finite domain.
The parametric equations are easier to enter into a calculator for graphing.
The parametric equations are easier to enter into a calculator for graphing.
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x = 1 + 6 cos t
y equals negative 2 plus 6 sine t.
What are the advantages of the parametric equations? Select all that apply.
(2 points)
Responses
The parametric equations show the direction in which the circle is drawn.
The parametric equations show the direction in which the circle is drawn.
The parametric equations describe a three-dimensional figure.
The parametric equations describe a three-dimensional figure.
For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
The parametric equations have a finite domain.
The parametric equations have a finite domain.
The parametric equations are easier to enter into a calculator for graphing.
The parametric equations are easier to enter into a calculator for graphing.
Skip to navigation
Answers
GPT-4o mini
Let's evaluate the advantages of the parametric equations for the circle compared to the Cartesian equation:
1. **The parametric equations show the direction in which the circle is drawn.**
- **True**: The parametric equations define \(x\) and \(y\) in terms of the parameter \(t\), which allows you to see how the points traverse the circle as \(t\) varies. This conveys the directionality (e.g., clockwise or counterclockwise) in which the circle is plotted.
2. **The parametric equations describe a three-dimensional figure.**
- **False**: Both the Cartesian and the parametric equations describe a two-dimensional circle in the \(xy\)-plane. The given parametric equations do not involve a third variable that would make it three-dimensional.
3. **For a given value of the independent variable, the parametric equations yield exactly one point on the graph.**
- **True**: For each value of \(t\), the parametric equations yield a unique point \((x, y)\) on the circle, unlike a single Cartesian equation where a given \(x\) can correspond to two \(y\) values (or vice versa).
4. **The parametric equations have a finite domain.**
- **False**: The parameter \(t\) can take any real number, which means the domain is generally infinite (though it can be restricted based on the context, such as \(t \in [0, 2\pi]\)).
5. **The parametric equations are easier to enter into a calculator for graphing.**
- **True**: Many graphing calculators can handle parametric equations directly, often making it easier to plot the curves compared to rearranging or solving Cartesian equations.
### Summary of Advantages:
- **The parametric equations show the direction in which the circle is drawn.**
- **For a given value of the independent variable, the parametric equations yield exactly one point on the graph.**
- **The parametric equations are easier to enter into a calculator for graphing.**
### Selected responses:
1. The parametric equations show the direction in which the circle is drawn.
2. For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
3. The parametric equations are easier to enter into a calculator for graphing.
1. **The parametric equations show the direction in which the circle is drawn.**
- **True**: The parametric equations define \(x\) and \(y\) in terms of the parameter \(t\), which allows you to see how the points traverse the circle as \(t\) varies. This conveys the directionality (e.g., clockwise or counterclockwise) in which the circle is plotted.
2. **The parametric equations describe a three-dimensional figure.**
- **False**: Both the Cartesian and the parametric equations describe a two-dimensional circle in the \(xy\)-plane. The given parametric equations do not involve a third variable that would make it three-dimensional.
3. **For a given value of the independent variable, the parametric equations yield exactly one point on the graph.**
- **True**: For each value of \(t\), the parametric equations yield a unique point \((x, y)\) on the circle, unlike a single Cartesian equation where a given \(x\) can correspond to two \(y\) values (or vice versa).
4. **The parametric equations have a finite domain.**
- **False**: The parameter \(t\) can take any real number, which means the domain is generally infinite (though it can be restricted based on the context, such as \(t \in [0, 2\pi]\)).
5. **The parametric equations are easier to enter into a calculator for graphing.**
- **True**: Many graphing calculators can handle parametric equations directly, often making it easier to plot the curves compared to rearranging or solving Cartesian equations.
### Summary of Advantages:
- **The parametric equations show the direction in which the circle is drawn.**
- **For a given value of the independent variable, the parametric equations yield exactly one point on the graph.**
- **The parametric equations are easier to enter into a calculator for graphing.**
### Selected responses:
1. The parametric equations show the direction in which the circle is drawn.
2. For a given value of the independent variable, the parametric equations yield exactly one point on the graph.
3. The parametric equations are easier to enter into a calculator for graphing.