The problem asks for the coordinates of a transformed point on the graph of the function $y = f(2x - 4)$, where the point $(4, -12)$ lies on the graph of $y = f(x)$.

Step-by-Step Solution:

1. Given: $(4, -12)$ is on the graph of $y = f(x)$. So, $f(4) = -12$.

2. Transformation: The transformation $y = f(2x - 4)$ involves a horizontal shift and stretch:

   • Shift: The expression inside the function, $2x - 4$, indicates a horizontal shift by 2 units (because $x - 2 = 0$ when $2x - 4 = 0$).
   • Stretch: The factor of 2 means the graph is horizontally compressed by a factor of $\frac{1}{2}$.

3. Find the new x-coordinate: To map the original point $(4, -12)$ onto the new function, we solve for $x$ in $2x - 4 = 4$ (since $f(4) = -12$): $2x - 4 = 4 \implies 2x = 8 \implies x = 4.$ So, the new x-coordinate is $x = 6$.

4. y-coordinate remains the same: Since $f(4) = -12$, the new y-coordinate for this point is still $-12$.

Therefore, the transformed point is $(6, -24)$.

Final Answer:

The correct transformed point is $(6, -24)$.

Do you need further details or have any questions?

Here are five related questions for further practice:

1. What happens to a point $(x, y)$ under the transformation $y = f(3x - 6)$?
2. If $y = f(x + 2)$, how does the graph of $f(x)$ shift?
3. How does the graph of $f(x)$ change when transformed into $y = f(-x)$?
4. What is the effect of the transformation $y = 2f(x)$ on the graph of $f(x)$?
5. How do you find the inverse of a horizontal stretch transformation?

Tip: Horizontal transformations are counterintuitive—they affect the graph by acting on the x-coordinate in the opposite way!