Question
x equals Start Root t End Root and y = 2t + 1
(1 point)
Responses
A graph on a coordinate plane shows the right side of a parabola.A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows on the parabola pointing up toward the right. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1 and a point at left-parenthesis 1.4 comma 5 right-parenthesis is labeled t = 2.
Image with alt text: A graph on a coordinate plane shows the right side of a parabola. A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows on the parabola pointing up toward the right. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1 and a point at left-parenthesis 1.4 comma 5 right-parenthesis is labeled t = 2.
A graph on a coordinate plane shows a parabola that opens upward.A graph on a coordinate plane shows a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing toward the right, down and up the parabola. A point at left-parenthesis negative 1 comma 3 right-parenthesis is labeled t equals negative 1, a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, and a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1.
Image with alt text: A graph on a coordinate plane shows a parabola that opens upward. A graph on a coordinate plane shows a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing toward the right, down and up the parabola. A point at left-parenthesis negative 1 comma 3 right-parenthesis is labeled t equals negative 1, a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, and a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1.
A graph on a coordinate plane shows the top side of a parabola that opens to the right.A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing up toward the right along the curve. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 3.
Image with alt text: A graph on a coordinate plane shows the top side of a parabola that opens to the right. A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing up toward the right along the curve. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 3.
A graph on a coordinate plane shows the top side of a parabola.A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing down toward the left along the curve. A point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 3.
Image with alt text: A graph on a coordinate plane shows the top side of a parabola. A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing down toward the left along the curve. A point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 3.
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(1 point)
Responses
A graph on a coordinate plane shows the right side of a parabola.A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows on the parabola pointing up toward the right. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1 and a point at left-parenthesis 1.4 comma 5 right-parenthesis is labeled t = 2.
Image with alt text: A graph on a coordinate plane shows the right side of a parabola. A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows on the parabola pointing up toward the right. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1 and a point at left-parenthesis 1.4 comma 5 right-parenthesis is labeled t = 2.
A graph on a coordinate plane shows a parabola that opens upward.A graph on a coordinate plane shows a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing toward the right, down and up the parabola. A point at left-parenthesis negative 1 comma 3 right-parenthesis is labeled t equals negative 1, a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, and a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1.
Image with alt text: A graph on a coordinate plane shows a parabola that opens upward. A graph on a coordinate plane shows a parabola that opens upward with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing toward the right, down and up the parabola. A point at left-parenthesis negative 1 comma 3 right-parenthesis is labeled t equals negative 1, a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, and a point at left-parenthesis 1 comma 3 right-parenthesis is labeled t = 1.
A graph on a coordinate plane shows the top side of a parabola that opens to the right.A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing up toward the right along the curve. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 3.
Image with alt text: A graph on a coordinate plane shows the top side of a parabola that opens to the right. A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing up toward the right along the curve. A point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 3.
A graph on a coordinate plane shows the top side of a parabola.A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing down toward the left along the curve. A point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 3.
Image with alt text: A graph on a coordinate plane shows the top side of a parabola. A graph on a coordinate plane shows the top side of a parabola that opens to the right with a vertex of left parenthesis 0 comma 1 right parenthesis. There are arrows pointing down toward the left along the curve. A point at left-parenthesis 9 comma 7 right-parenthesis is labeled t = 0, a point at left-parenthesis 4 comma 5 right-parenthesis is labeled t = 2 and a point at left-parenthesis 0 comma 1 right-parenthesis is labeled t = 3.
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Answers
GPT-4o mini
To analyze the given parametric equations \(x = \sqrt{t}\) and \(y = 2t + 1\), we can rewrite them in a Cartesian format to determine the nature of the graph they represent.
1. **Eliminate the Parameter \(t\)**:
From the equation \(x = \sqrt{t}\), we can express \(t\) in terms of \(x\):
\[
t = x^2
\]
2. **Substitute \(t\) into the \(y\) equation**:
Substitute \(t = x^2\) into the second equation \(y = 2t + 1\):
\[
y = 2(x^2) + 1 = 2x^2 + 1
\]
This equation \(y = 2x^2 + 1\) represents a parabola that opens upward with its vertex at the point \((0, 1)\).
3. **Analyze the Properties**:
- The vertex \( (0, 1) \) is the minimum point of the parabola.
- As \(x\) increases or decreases from 0, \(y\) increases, consistent with the shape of a parabola that opens upwards.
4. **Identify the Graph**:
- The parabola opens upward, and the vertex is at \( (0, 1) \).
- For increasing values of \(t\), the points on the graph start from the vertex and move upward as seen from the equations.
Given this analysis, the correct description matches the scenario described in the first response option:
- **A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of \((0, 1)\) ... labeled appropriately for points corresponding to values of \(t\)**.
### Conclusion:
The graph that best fits the equations given and their analysis is the **first option** described. The parabola opens upwards with the vertex at \((0,1)\).
1. **Eliminate the Parameter \(t\)**:
From the equation \(x = \sqrt{t}\), we can express \(t\) in terms of \(x\):
\[
t = x^2
\]
2. **Substitute \(t\) into the \(y\) equation**:
Substitute \(t = x^2\) into the second equation \(y = 2t + 1\):
\[
y = 2(x^2) + 1 = 2x^2 + 1
\]
This equation \(y = 2x^2 + 1\) represents a parabola that opens upward with its vertex at the point \((0, 1)\).
3. **Analyze the Properties**:
- The vertex \( (0, 1) \) is the minimum point of the parabola.
- As \(x\) increases or decreases from 0, \(y\) increases, consistent with the shape of a parabola that opens upwards.
4. **Identify the Graph**:
- The parabola opens upward, and the vertex is at \( (0, 1) \).
- For increasing values of \(t\), the points on the graph start from the vertex and move upward as seen from the equations.
Given this analysis, the correct description matches the scenario described in the first response option:
- **A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of \((0, 1)\) ... labeled appropriately for points corresponding to values of \(t\)**.
### Conclusion:
The graph that best fits the equations given and their analysis is the **first option** described. The parabola opens upwards with the vertex at \((0,1)\).