The image contains a diagram of triangles $ABC$ and $A'B'C'$ drawn on a coordinate plane. The problem asks for the equation of the line connecting certain points.

Here is how you can solve this:

Step 1: Identifying coordinates

From the image:

• $A = (0, 4)$
• $B = (2, 2)$
• $C = (-4, -4)$

Step 2: Finding the equation of the line

To determine the equation of the line through points $A$ and $B$:

1. Slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $A(0, 4)$ and $B(2, 2)$: $m = \frac{2 - 4}{2 - 0} = \frac{-2}{2} = -1$

2. Equation of the line in point-slope form is: $y - y_1 = m(x - x_1)$ Using point $A(0, 4)$ and $m = -1$: $y - 4 = -1(x - 0)$ Simplify: $y = -x + 4$

Final Answer:

The equation of the line connecting points $A$ and $B$ is: $y = -x + 4$

Let me know if you'd like me to clarify this or explore further! Here are some related questions to build understanding:

1. How do you calculate the slope of a line between any two points?
2. What is the general form of the equation of a line, and how does it relate to point-slope form?
3. How can you find the intersection of two lines given their equations?
4. What transformations are applied to move triangle $ABC$ to triangle $A'B'C'$?
5. How do you determine if two lines are parallel or perpendicular?

Tip: Always double-check coordinates and calculations when working with graphs on a grid.