The given problem asks for the transformation of the linear function $y = x + 7$ when it is rotated clockwise by $90^\circ$ about the origin.

Step-by-step solution:

1. Original Line Equation: The line equation is $y = x + 7$. In slope-intercept form, the slope $m$ of this line is 1 (since $y = mx + c$, where $m = 1$ and $c = 7$).

2. Clockwise Rotation by $90^\circ$:

   • When a line with slope $m$ is rotated by $90^\circ$ clockwise about the origin, its new slope becomes the negative reciprocal of $m$, i.e., $m_{\text{new}} = -\frac{1}{m}$.
   • In this case, since $m = 1$, the new slope after a $90^\circ$ clockwise rotation will be $m_{\text{new}} = -1$.

3. New Line Equation:

   • The new equation of the line will have the form $y = -x + c$. To find $c$, the line must pass through the same point on the original line that was rotated.
   • The line intersects the y-axis at $(0, 7)$. After rotating this point $90^\circ$ clockwise, the new coordinates of the point are $(7, 0)$.

4. Final Equation:

   • Using the point $(7, 0)$, substitute it into the new equation $y = -x + c$: $0 = -7 + c \implies c = 7$
   • Thus, the new equation of the line is $y = -x + 7$.

Conclusion:

The equation of the line after a $90^\circ$ clockwise rotation is $y = -x + 7$.

Would you like more details or have any questions?

Here are five related questions to explore further:

1. What happens when the line is rotated $90^\circ$ counterclockwise instead?
2. How do you rotate a line around a different point other than the origin?
3. What is the geometric meaning of rotating a line about the origin?
4. How would the transformation change if the rotation angle were $180^\circ$?
5. Can you rotate other types of curves, like parabolas, in the same way?

Tip: Rotating lines involves changing their slope, but always remember that rotations around the origin also involve changing the coordinates of points on the line!