https://file.aiquickdraw.com/mathbot/file/1727669779737opagyfd6.jpg

The problem asks for the transformation of the function \( f(x) = |x| \) to a new function \( g(x) \), following these transformations:
1. **Horizontally stretching** by a factor of 7,
2. **Reflecting across the x-axis**, and
3. **Shifting up by 6 units**.

Let's break this down step by step:

1. **Horizontally stretching by a factor of 7**: This means that the x-values will be scaled by \( \frac{1}{7} \). So, the horizontal stretching transformation results in \( f\left(\frac{x}{7}\right) \), which becomes \( \left|\frac{x}{7}\right| \).

2. **Reflecting across the x-axis**: Reflecting the graph across the x-axis introduces a negative sign in front of the function, so the transformation becomes \( -\left|\frac{x}{7}\right| \).

3. **Shifting up by 6 units**: To shift the graph upwards by 6 units, we add 6 to the function, resulting in \( -\left|\frac{x}{7}\right| + 6 \).

Thus, the final transformed function \( g(x) \) is:
\[
g(x) = -\left|\frac{x}{7}\right| + 6
\]

This matches the formula provided in the image: 
\[
g(x) = -\left(\frac{x}{7}\right) + 6
\]

Would you like further clarification on any part of this transformation process? Here are some follow-up questions you might consider:

1. What does a horizontal stretch do to the graph of a function in general?
2. How does reflecting across the x-axis affect the graph of any function?
3. Can you describe how shifting a graph up or down changes its appearance?
4. Why is the horizontal stretch by 7 represented as \( \frac{x}{7} \)?
5. What would happen if we shifted the graph down instead of up by 6 units?

**Tip:** Always apply transformations in the correct order (stretches/compressions, reflections, and then shifts).

Find a formula for a function g(x) whose graph is obtained from f(x) = |x| by horizontally stretching by a factor of 7, reflecting across the x-axis, and shifting up 6 units.

Learn how to transform the absolute value function f(x) = |x| by applying a horizontal stretch, reflection across the x-axis, and vertical shift. The final function g(x) = -|x/7| + 6 demonstrates these steps clearly, suitable for students in Grades 10-12.

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