Question
A curve is described by the following parametric equations:
x equals 3 plus t line break y equals t squared minus 4
Which statement best describes the curve?
(1 point)
Responses
The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t .
The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from right to left for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis 3 comma negative 4 right parenthesis and is traced from right to left for increasing values of t .
The curve is a parabola with a vertex at left parenthesis negative 3 comma 4 right parenthesis and is traced from left to right for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis negative 3 comma 4 right parenthesis and is traced from left to right for increasing values of t .
The curve is a parabola with a vertex at left parenthesis negative 3 comma 4 right parenthesis and is traced from right to left for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis negative 3 comma 4 right parenthesis and is traced from right to left for increasing values of t .
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x equals 3 plus t line break y equals t squared minus 4
Which statement best describes the curve?
(1 point)
Responses
The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t .
The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from right to left for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis 3 comma negative 4 right parenthesis and is traced from right to left for increasing values of t .
The curve is a parabola with a vertex at left parenthesis negative 3 comma 4 right parenthesis and is traced from left to right for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis negative 3 comma 4 right parenthesis and is traced from left to right for increasing values of t .
The curve is a parabola with a vertex at left parenthesis negative 3 comma 4 right parenthesis and is traced from right to left for increasing values of t.
The curve is a parabola with a vertex at Image with alt text: left parenthesis negative 3 comma 4 right parenthesis and is traced from right to left for increasing values of t .
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Answers
GPT-4o mini
To analyze the given parametric equations:
\[
x = 3 + t
\]
\[
y = t^2 - 4
\]
1. **Identifying the Vertex:**
- The \(x\) equation indicates a horizontal translation of 3 from the origin.
- The \(y\) equation, \(y = t^2 - 4\), represents a standard parabola that opens upwards with its vertex at \( (0, -4) \).
- Since we have \(x = 3 + t\), we can rewrite \(t\) in terms of \(x\):
\[
t = x - 3
\]
- Substitute \(t\) back into the \(y\) equation:
\[
y = (x - 3)^2 - 4
\]
- This gives us the vertex of the parabola at \( (3, -4) \).
2. **Direction of Tracing:**
- As \(t\) increases from negative to positive values, the corresponding \(x\) values are:
- When \(t = 0\), \(x = 3\).
- When \(t\) increases, \(x\) also increases.
- The \(y\) values will increase as \(t\) increases (since \(t^2\) is always non-negative and increases as \(t\) moves away from zero in both the positive and negative directions).
Thus, the curve is a parabola with a vertex at \( (3, -4) \) and is traced from left to right for increasing values of \(t\).
**Correct Statement:**
The curve is a parabola with a vertex at \( (3, -4) \) and is traced from left to right for increasing values of \(t\).
\[
x = 3 + t
\]
\[
y = t^2 - 4
\]
1. **Identifying the Vertex:**
- The \(x\) equation indicates a horizontal translation of 3 from the origin.
- The \(y\) equation, \(y = t^2 - 4\), represents a standard parabola that opens upwards with its vertex at \( (0, -4) \).
- Since we have \(x = 3 + t\), we can rewrite \(t\) in terms of \(x\):
\[
t = x - 3
\]
- Substitute \(t\) back into the \(y\) equation:
\[
y = (x - 3)^2 - 4
\]
- This gives us the vertex of the parabola at \( (3, -4) \).
2. **Direction of Tracing:**
- As \(t\) increases from negative to positive values, the corresponding \(x\) values are:
- When \(t = 0\), \(x = 3\).
- When \(t\) increases, \(x\) also increases.
- The \(y\) values will increase as \(t\) increases (since \(t^2\) is always non-negative and increases as \(t\) moves away from zero in both the positive and negative directions).
Thus, the curve is a parabola with a vertex at \( (3, -4) \) and is traced from left to right for increasing values of \(t\).
**Correct Statement:**
The curve is a parabola with a vertex at \( (3, -4) \) and is traced from left to right for increasing values of \(t\).