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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.

Describe a function g(x) in terms of f(x) if the graph of g is obtained by reflecting the graph of f about the x-axis and if it is horizontally stretched by a factor of 5 when compared to the graph of f. The equation is g(x) = Af(Bx) + C where A = ?, B = ?, C = ?.

Learn how to transform a function with reflections and horizontal stretching. This problem involves describing g(x) in terms of f(x), reflecting the graph over the x-axis, and horizontally stretching by a factor of 5. Suitable for Grades 10-12 students.

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