https://file.aiquickdraw.com/mathbot/file/172697451171909cjrlnh.jpg

To find a parameterization for the curve shown in the image, we need to follow these steps:

### 1. Identify the two points on the curve:
- The curve is a straight line between two points.
- The starting point is \((0, 1)\), and the ending point is \((2, 3)\).

### 2. Use the parametric form for a line segment:
A line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be parameterized as:
\[
\vec{r}(t) = (1 - t)\vec{r}_0 + t\vec{r}_1
\]
where \( \vec{r}_0 = (x_1, y_1) \) and \( \vec{r}_1 = (x_2, y_2) \), and \( t \) is the parameter ranging from 0 to 1.

### 3. Apply the points:
For this line segment, \( \vec{r}_0 = (0, 1) \) and \( \vec{r}_1 = (2, 3) \). The parameterized equation is:
\[
\vec{r}(t) = (1 - t)(0, 1) + t(2, 3)
\]
Breaking this down for the \(x\)- and \(y\)-coordinates, we get:
\[
x(t) = (1 - t)(0) + t(2) = 2t
\]
\[
y(t) = (1 - t)(1) + t(3) = 1 + 2t
\]

### 4. Define the parameter \(t\):
The parameter \(t\) typically varies from 0 to 1, as it traces the line segment from the starting point \((0, 1)\) when \(t = 0\) to the ending point \((2, 3)\) when \(t = 1\).

### Final Parameterization:
\[
\vec{r}(t) = \left( 2t, 1 + 2t \right), \quad 0 \leq t \leq 1
\]

So the parameterization for the curve is:
\[
\vec{r}(t) = \left( 2t, 1 + 2t \right), \quad 0 \leq t \leq 1
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you parameterize curves that are not straight lines?
2. What is the general form of a line in parametric equations?
3. How can you extend this to three dimensions for space curves?
4. How do parametric equations relate to vector-valued functions?
5. What is the difference between Cartesian and parametric representations?

**Tip:** The parameter \(t\) often represents time or another progression variable, which makes parametric equations versatile in physics and animations.

Find a parameterization for the curve shown in the figure below.

Learn how to find the parameterization of a line segment between two points using vector equations and parametric forms. This tutorial is suitable for advanced high school students and undergraduates, covering key concepts in linear algebra and vector mathematics.

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