To identify the curve represented by the parametric equations \( x = 2 + 3 \sin t \) and \( y = 1 - \frac{1}{2} \cos t \), we can analyze each equation.
Step 1: Rewrite the equations in terms of sine and cosine
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For \( x \): \[ x - 2 = 3 \sin t \quad \Rightarrow \quad \sin t = \frac{x - 2}{3} \]
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For \( y \): \[ y - 1 = -\frac{1}{2} \cos t \quad \Rightarrow \quad \cos t = -2(y - 1) = 2 - 2y \]
Step 2: Use the identity \( \sin^2 t + \cos^2 t = 1 \)
We can substitute the expressions for \(\sin t\) and \(\cos t\) into this identity:
\[ \left(\frac{x - 2}{3}\right)^2 + (2 - 2y)^2 = 1 \]
Expanding and simplifying:
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Calculate \(\left(\frac{x - 2}{3}\right)^2\): \[ \frac{(x - 2)^2}{9} \]
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Calculate \((2 - 2y)^2\): \[ 4(1 - y)^2 = 4(1 - 2y + y^2) = 4 - 8y + 4y^2 \]
Combining these we get: \[ \frac{(x - 2)^2}{9} + \frac{4 - 8y + 4y^2}{4} = 1 \]
Step 3: Multiply through by 36 (the least common multiple of 9 and 4):
\[ 4(x - 2)^2 + 9(4 - 8y + 4y^2) = 36 \] \[ 4(x - 2)^2 + 36 - 72y + 36y^2 = 36 \] \[ 4(x - 2)^2 + 36y^2 - 72y = 0 \]
Step 4: Rearrange to match the form of an ellipse
The equation can be rearranged to resemble the standard form of an ellipse by completing the square, but we can already identify the important characteristics:
- The center of this ellipse is at \( (2, 1) \).
- The semi-major axis corresponds to the coefficient of \( \sin t \) and \( 3 \) indicates a horizontal stretch of 3.
- The semi-minor axis corresponds to the coefficient of \( \cos t \) which is \( \frac{1}{2} \times 2 = 1 \).
Step 5: Determine the orientation
Since \( x \) is expressed in terms of \( \sin t \) (which varies from -1 to 1), the major axis of the ellipse will be horizontal.
Conclusion
Putting it all together, Janet is drawing a horizontal ellipse centered at \((2, 1)\) with a major axis length of 6 (from \(-3\) to \(3\) around the center) and a minor axis length of 1.
The direction traced by the curve is counterclockwise as \(t\) increases, because:
- \( \sin t \) increases from 0 to 1 to 0 to -1 and back to 0 in the first cycle, resulting in a counterclockwise motion in the xy-plane.
Thus, the correct answer is:
1. 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.