To analyze the given parametric equations:
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Parametric Equations:
- \( x = 3 + t \)
- \( y = t^2 - 4 \)
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Finding the Vertex:
- The equation for \( y \) can be rewritten as \( y = (t^2 - 4) \). This is a standard upward-opening parabola with a vertex given by the value of \( t \) that minimizes \( y \).
- The minimum value of \( t^2 \) occurs at \( t = 0 \), where \( y = 0^2 - 4 = -4 \).
- To find \( x \) at this vertex point: \[ x = 3 + 0 = 3 \]
- So, the vertex of the parabola is at \( (3, -4) \).
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Tracing the Curve:
- As \( t \) increases, \( x \) increases since \( x = 3 + t \).
- The corresponding values of \( y \) for increasing \( t \):
- At \( t = -1 \), \( (x, y) = (3 - 1, 1 - 4) = (2, -3) \)
- At \( t = 0 \), \( (x, y) = (3, -4) \)
- At \( t = 1 \), \( (x, y) = (3 + 1, 1 - 4) = (4, -3) \)
- Thus, as \( t \) goes from negative values to positive values, the points on the curve move from left to right.
Based on this analysis, the best description of the curve is:
The curve is a parabola with a vertex at (3, -4) and is traced from left to right for increasing values of t.