To translate a shape using the vector $(-2, 3)$, you will shift each point of the shape 2 units to the left and 3 units upward.

When translating a shape, the entire figure is moved without any rotation or resizing. The vector $(-2, 3)$ provides precise instructions for how to move each point of the shape: the first component, $-2$, indicates a horizontal shift of 2 units to the left, while the second component, $3$, signifies a vertical shift of 3 units upward.

For instance, consider a point located at $(4, 1)$ on your shape. To translate this point, you will move it to $(2, 4)$. This is achieved by subtracting $2$ from the x-coordinate: $4 - 2 = 2$, and adding $3$ to the y-coordinate: $1 + 3 = 4$. You will apply this process to every point on the shape to complete the translation.

Let’s examine a triangle with vertices at $(1, 2)$, $(3, 4)$, and $(5, 6)$. To translate this triangle using the vector $(-2, 3)$, you will adjust each vertex accordingly: $(1, 2)$ becomes $(-1, 5)$, $(3, 4)$ changes to $(1, 7)$, and $(5, 6)$ moves to $(3, 9)$. While the shape of the triangle remains unchanged, its position in the coordinate plane is altered.

Translating shapes is a fundamental concept in geometry, crucial for understanding the movement of figures within a coordinate system. It’s akin to providing specific directions to each point on the shape, guiding them to their new locations. Mastering this skill is beneficial in various areas of mathematics and lays a strong foundation for more advanced topics.