To solve this problem, you are given a function $f(x)$, and you need to describe the transformation to get the function $g(x)$, which is reflected across the x-axis and horizontally stretched by a factor of 5. The form of $g(x)$ is given as:

$g(x) = A f(Bx) + C$

Let's break down the transformations:

1. Reflection across the x-axis: Reflecting a graph across the x-axis changes the sign of the function's output. This is achieved by multiplying the function by $-1$. Therefore, $A = -1$.

2. Horizontal stretching: A horizontal stretch by a factor of 5 means that the graph of the function becomes "wider." To apply this stretch, we modify the input by multiplying $x$ by a factor of $\frac{1}{5}$. Therefore, $B = \frac{1}{5}$.

3. Vertical shift: There is no mention of any vertical shift, so $C = 0$.

Thus, the values of $A$, $B$, and $C$ are:

$A = -1, \quad B = \frac{1}{5}, \quad C = 0$

Therefore, the function $g(x)$ can be written as:

$g(x) = -f\left(\frac{1}{5}x\right)$

Would you like more details on this? Here are some related questions to expand understanding:

1. What would happen if the function was vertically stretched instead of horizontally stretched?
2. How would you modify the function for a reflection over the y-axis instead?
3. If there were a vertical shift of 3 units upwards, how would that affect $C$?
4. How does a horizontal compression differ from a horizontal stretch in terms of the value of $B$?
5. Can you explain how multiple transformations affect the overall function graphically?

Tip: Always apply transformations in the correct order (horizontal, reflections, stretches, and vertical) for consistent results when graphing.