Question

Which is the graph of the following parametric equation?

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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Answer 1

To determine the graph of the parametric equations \( x = \sqrt{t} \) and \( y = 2t + 1 \), let's first analyze these equations.

1. **Finding the relationship between \( x \) and \( y \)**:
- From \( x = \sqrt{t} \), we can solve for \( t \):
\[
t = x^2
\]
- Substituting \( t \) back into the equation for \( y \):
\[
y = 2t + 1 = 2(x^2) + 1 = 2x^2 + 1
\]

This shows that the relationship between \( y \) and \( x \) is a quadratic function, specifically a parabola that opens upwards with the vertex at \( (0, 1) \).

2. **Analyzing the parameterization**:
- As \( t \) increases from \( 0 \) onward, \( x \) will take values starting from \( 0 \) and increase (since \( x = \sqrt{t} \) for \( t \geq 0 \)).
- At \( t = 0 \), \( (x, y) = (0, 1) \).
- At \( t = 1 \), \( (x, y) = (1, 3) \).
- At \( t = 2 \), \( (x, y) = (1.414, 5) \) approximately.

Since \( x \) starts at \( 0 \) and increases, and \( y \) also increases as \( t \) increases, the graph will be the right side of the parabola, opening upwards, beginning from \( (0, 1) \) and extending to the right.

3. **Identifying the correct graph**:
Based on the characteristics derived, the graph depicting the right side of a parabola that opens upward with the vertex at \( (0, 1) \) matches the provided options. Specifically, it should have the following points:
- At \( t = 0 \): \( (0, 1) \)
- At \( t = 1 \): \( (1, 3) \)
- At \( t = 2 \): Approximately \( (1.414, 5) \)

Therefore, the correct response is:

**A graph on a coordinate plane shows the right side of a parabola that opens upward with a vertex of \( (0, 1) \)**.