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.