The problem is asking to find the image of the line $y + 2x - 4 = 0$ when rotated by $-90^\circ$ around the origin $(0, 0)$.

Steps to solve:

1. Rewrite the line equation in slope-intercept form: $y = -2x + 4$ This has a slope of $-2$ and a y-intercept of 4.

2. Understand rotation of points: When a point $(x, y)$ is rotated counterclockwise by $-90^\circ$ (which is clockwise), the transformation is given by: $(x', y') = (y, -x)$

3. Transform the line under rotation: To rotate the entire line, substitute each point $(x, y)$ on the line and apply the transformation.

   Since the original equation is $y = -2x + 4$, substitute into the rotation formula:

   • The original line: $y = -2x + 4$
   • Rotation by $-90^\circ$: Each point on the line $(x, y)$ transforms to $(y, -x)$.

4. Apply the transformation:

   • The new $x'$ will be the old $y$, and the new $y'$ will be the negative of the old $x$.

   This leads to the transformed equation: $x' = -2y' + 4$ Rearranging for standard linear form, the equation becomes: $x' + 2y' - 4 = 0$

Thus, after a $-90^\circ$ rotation around the origin, the image of the line is: $x + 2y - 4 = 0$

Would you like any further clarifications or details on this solution?

Here are 5 related questions to expand on the topic:

1. How do rotations affect the slopes of lines in general?
2. What is the transformation matrix for a $+90^\circ$ rotation around the origin?
3. How would the line change if rotated by $180^\circ$ instead?
4. Can the rotation of more complex curves (e.g., circles or parabolas) be handled similarly?
5. How does translation affect a line differently compared to rotation?

Tip: Understanding transformations geometrically can make it easier to visualize how shapes and lines behave under rotation or translation.