The graph of `{:{(x=sin^(2)t),(y=2 cos t):}`

Options

A. straight line
B. line segment
C. parabola
D. portion of a parabola

Explanation

To solve the problem of finding the graph of the parametric equations given by \( x = \sin^2 t \) and \( y = 2 \cos t \), we will eliminate the parameter \( t \) to express \( x \) in terms of \( y \) and identify the resulting curve.

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ x = \sin^2 t \]
\[ y = 2 \cos t \]

2. **Express \( \cos t \) in terms of \( y \):**
From the equation for \( y \):
\[ \cos t = \frac{y}{2} \]

3. **Use the Pythagorean identity:**
We know that:
\[ \sin^2 t + \cos^2 t = 1 \]
Substitute \( \cos t \) from the previous step:
\[ \sin^2 t + \left(\frac{y}{2}\right)^2 = 1 \]

4. **Substitute \( \sin^2 t \) with \( x \):**
Since \( x = \sin^2 t \), we can substitute:
\[ x + \left(\frac{y}{2}\right)^2 = 1 \]

5. **Simplify the equation:**
Expanding the equation gives:
\[ x + \frac{y^2}{4} = 1 \]
Rearranging it, we get:
\[ \frac{y^2}{4} = 1 - x \]

6. **Multiply through by 4 to eliminate the fraction:**
\[ y^2 = 4(1 - x) \]

7. **Rearranging gives us the standard form of a parabola:**
\[ y^2 = -4x + 4 \]
This can be rewritten as:
\[ y^2 = 4(1 - x) \]

8. **Identify the characteristics of the parabola:**
The equation \( y^2 = 4(1 - x) \) represents a parabola that opens to the left with its vertex at \( (1, 0) \).

9. **Determine the range of \( x \):**
Since \( x = \sin^2 t \), and \( \sin^2 t \) varies from \( 0 \) to \( 1 \), we have:
\[ 0 \leq x \leq 1 \]

10. **Conclusion about the graph:**
The graph of the parametric equations is a portion of the parabola \( y^2 = 4(1 - x) \) that is limited to the values of \( x \) from \( 0 \) to \( 1 \).