The question you uploaded asks:

"If (0, 8) is a point on the graph of $y = f(x)$, which of the following must be on the graph of $y = f(-x)$?"

Solution:

1. Understanding the relationship between $f(x)$ and $f(-x)$:

   • The transformation $y = f(-x)$ is a horizontal reflection of the graph of $y = f(x)$ across the y-axis.

2. Given point:

   • $(0, 8)$ is on the graph of $y = f(x)$, meaning $f(0) = 8$.

3. Reflecting over the y-axis:

   • When we reflect a point $(a, b)$ across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same.
   • In this case, the point $(0, 8)$ is on $f(x)$, and reflecting this point across the y-axis results in no change because the x-coordinate is 0. So, the point remains $(0, 8)$.

Therefore, (0, 8) must also be on the graph of $y = f(-x)$.

The correct answer is:

• (0, 8).

Would you like more details or have any questions?

5 related questions:

1. What kind of transformation is $f(-x)$ compared to $f(x)$?
2. How do vertical and horizontal transformations affect graphs of functions?
3. Can we apply this reflection concept to non-symmetric functions?
4. What happens to the graph of $f(x)$ if the transformation is $y = -f(x)$?
5. How does the graph of $y = f(x + 2)$ differ from $y = f(x)$?

Tip:

When reflecting a graph horizontally (via $f(-x)$), only the x-coordinates change, while the y-coordinates stay the same.