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.