Question
Janet is drawing the path formed by the parametric equations x = 2 + 3 sin t and y equals 1 minus start fraction 1 over 2 end fraction cosine t. Which curve did she draw?
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Responses
A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
Image with alt text: A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
Image with alt text: A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
Image with alt text: A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
Image with alt text: A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
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(1 point)
Responses
A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
Image with alt text: A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
Image with alt text: A horizontal ellipse is graphed on a coordinate plane centered at left-parenthesis 2 comma 1 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
Image with alt text: A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.
A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
Image with alt text: A vertical ellipse is graphed on a coordinate plane centered at left-parenthesis 1 comma 2 right-parenthesis with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a clockwise direction.
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Answers
Answered by
GPT-4o mini
To determine the type of curve formed by the parametric equations \( x = 2 + 3 \sin t \) and \( y = 1 - \frac{1}{2} \cos t \), we can analyze the equations.
1. **Look at the equations**:
- The \( x \) equation is \( x = 2 + 3 \sin t \). This indicates that \( x \) varies based on the sine function, which has a range that stretches from \( 2 - 3 \) to \( 2 + 3 \) (i.e., from \(-1\) to \(5\)).
- The \( y \) equation is \( y = 1 - \frac{1}{2} \cos t \). This indicates that \( y \) varies based on the cosine function, ranging from \( 1 - \frac{1}{2}(-1) \) to \( 1 - \frac{1}{2}(1) \) (i.e., from \(1.5\) to \(0.5\)).
2. **Form of the equations**:
- A horizontal ellipse in parametric form can generally be represented as:
\[
x = h + a \cos t
\]
\[
y = k + b \sin t
\]
- In this case, \( h = 2 \), \( a = 3 \), \( k = 1 \), and \( b = \frac{1}{2} \).
3. **Determine the dimensions**:
- From \( x = h + a \sin t \), the length of the semi-major axis (along the x-axis) is \( 3 \), making the total length of the major axis \( 6 \) (i.e., \(2a = 6\)).
- From \( y = k - \frac{1}{2} \cos t \), the semi-minor axis (along the y-axis) has a length of \( \frac{1}{2} \), making the total length of the minor axis \( 1 \) (i.e., \(2b = 1\)).
4. **Centered Location**:
- The center of the ellipse is at the point \( (2, 1) \).
5. **Direction of traversal**:
- As \( t \) increases from \(0\) to \(2\pi\), \( \sin t\) varies from \(0\) to \(1\) to \(0\) to \(-1\) and back, while \( \cos t\) varies from \(1\) to \(0\) to \(-1\) to \(0\) back to \(1\). This corresponds to moving counterclockwise around the ellipse.
From this analysis, we can conclude that the curve Janet drew is:
**A horizontal ellipse is graphed on a coordinate plane centered at \( (2, 1) \) with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.**
1. **Look at the equations**:
- The \( x \) equation is \( x = 2 + 3 \sin t \). This indicates that \( x \) varies based on the sine function, which has a range that stretches from \( 2 - 3 \) to \( 2 + 3 \) (i.e., from \(-1\) to \(5\)).
- The \( y \) equation is \( y = 1 - \frac{1}{2} \cos t \). This indicates that \( y \) varies based on the cosine function, ranging from \( 1 - \frac{1}{2}(-1) \) to \( 1 - \frac{1}{2}(1) \) (i.e., from \(1.5\) to \(0.5\)).
2. **Form of the equations**:
- A horizontal ellipse in parametric form can generally be represented as:
\[
x = h + a \cos t
\]
\[
y = k + b \sin t
\]
- In this case, \( h = 2 \), \( a = 3 \), \( k = 1 \), and \( b = \frac{1}{2} \).
3. **Determine the dimensions**:
- From \( x = h + a \sin t \), the length of the semi-major axis (along the x-axis) is \( 3 \), making the total length of the major axis \( 6 \) (i.e., \(2a = 6\)).
- From \( y = k - \frac{1}{2} \cos t \), the semi-minor axis (along the y-axis) has a length of \( \frac{1}{2} \), making the total length of the minor axis \( 1 \) (i.e., \(2b = 1\)).
4. **Centered Location**:
- The center of the ellipse is at the point \( (2, 1) \).
5. **Direction of traversal**:
- As \( t \) increases from \(0\) to \(2\pi\), \( \sin t\) varies from \(0\) to \(1\) to \(0\) to \(-1\) and back, while \( \cos t\) varies from \(1\) to \(0\) to \(-1\) to \(0\) back to \(1\). This corresponds to moving counterclockwise around the ellipse.
From this analysis, we can conclude that the curve Janet drew is:
**A horizontal ellipse is graphed on a coordinate plane centered at \( (2, 1) \) with a major axis length of 6 and a minor axis length of 1. Arrows on the curve point in a counterclockwise direction.**
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