https://file.aiquickdraw.com/mathbot/file/1726280966814ibfjegxs.jpg

Let's break down the two problems presented in the image.

### Problem 1:
You are asked to write an equation for the function \( f(x) = |x| \), but shifted **6 units to the left** and **8 units down**. 

- Shifting a function horizontally by \( h \) units to the left is done by replacing \( x \) with \( x + h \).
- Shifting a function vertically by \( k \) units down is done by subtracting \( k \) from the function.

For this function:
- The shift of **6 units to the left** means \( x + 6 \).
- The shift of **8 units down** means subtracting 8.

Thus, the equation becomes:
\[
g(x) = |x + 6| - 8
\]

### Problem 2:
You are asked to write an equation based on the graph of \( f(x) = |x| \).

Looking at the graph (though it's hard to see specific details), it seems like a standard absolute value function that might be shifted. To write the correct equation, you would observe:
- Any horizontal or vertical shifts,
- Any changes in the slope or reflection, and
- The vertex's coordinates.

However, with only a partial image of the graph, I'd need more details to provide the specific equation.

Would you like a deeper explanation for these transformations, or should I help you analyze the second part further? Here are some related questions to consider:

1. What happens if you shift the function 6 units to the right instead of left?
2. How would you adjust the equation if the graph is reflected over the x-axis?
3. What if the function was shifted up instead of down—how would the equation change?
4. How do you describe a vertical stretch or compression of the absolute value function?
5. How would the equation change if the vertex of the absolute value graph was shifted to a different point?

**Tip:** Always keep track of the transformations by modifying \( x \) for horizontal shifts and applying operations directly to the function for vertical shifts.

Write an equation for the function f(x) = |x|, but shifted 6 units to the left and 8 units down. Then, use the graph of f(x) = |x| to write an equation for another shifted function based on the graph.

Learn how to write an equation for the absolute value function f(x) = |x|, shifted 6 units to the left and 8 units down. This example covers horizontal and vertical function shifts, useful for students in Grades 8-10 studying algebraic transformations.

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