To solve the problem of graphing the image of the triangle $\triangle QRS$ after a dilation with a scale factor of 4 centered at the origin, we need to follow these steps:

Step-by-Step Solution:

1. Identify the coordinates of points $Q$, $R$, and $S$:

   • From the image, the coordinates are approximately:
     • $Q(-6, 2)$
     • $R(4, 2)$
     • $S(-2, -2)$

2. Apply the dilation transformation:

   • The formula for dilation with a scale factor $k$ centered at the origin is: $(x, y) \to (kx, ky)$
   • Here, the scale factor $k = 4$.

3. Calculate the new coordinates for each point:

   • For point $Q(-6, 2)$: $Q' = (4 \cdot (-6), 4 \cdot 2) = (-24, 8)$
   • For point $R(4, 2)$: $R' = (4 \cdot 4, 4 \cdot 2) = (16, 8)$
   • For point $S(-2, -2)$: $S' = (4 \cdot (-2), 4 \cdot (-2)) = (-8, -8)$

4. Plot the new points $Q'$, $R'$, and $S'$ on the graph:

   • $Q'(-24, 8)$
   • $R'(16, 8)$
   • $S'(-8, -8)$

5. Draw the image of $\triangle Q'R'S'$ on the coordinate plane.

The new triangle will be much larger, with its vertices at the new calculated coordinates. Make sure to use the same coordinate grid to properly plot these points.

Would you like further details or have any questions? Here are some related questions that might interest you:

1. What is the formula for dilating a point not centered at the origin?
2. How do different scale factors affect the size and orientation of a figure?
3. How can you find the center of dilation when it's not given?
4. What are some practical applications of dilation in real-world scenarios?
5. How does dilation differ from other transformations like translation or rotation?

Tip: When working with dilations, always pay close attention to the scale factor and the center of dilation, as they determine the direction and magnitude of the transformation.