To convert the graph of $f(x) = -5|x - 7| - 7$ into the graph of $g(x) = -5|x + 1| - 7$, you apply a horizontal translation.

Transformation Details:

• The coefficient of the absolute value term and the constant term outside the absolute value (i.e., $-7$) are the same in both functions, so the vertical position and shape of the graph remain unchanged.
• The difference lies in the expression inside the absolute value: $|x - 7|$ in $f(x)$ and $|x + 1|$ in $g(x)$.

Horizontal Translation:

• $f(x)$ has $x - 7$ inside the absolute value, which means the graph of $f(x)$ is shifted 7 units to the right from the standard $|x|$ graph.
• $g(x)$ has $x + 1$, equivalent to $x - (-1)$, meaning the graph is shifted 1 unit to the left from the standard $|x|$ graph.

To move from $x - 7$ to $x + 1$, you translate the graph 8 units to the left.

Would you like more details or have any questions?

Related Questions:

1. What is the impact of shifting the graph horizontally on the vertex of the absolute value function?
2. How would you describe the transformation if the graph of $f(x)$ were shifted 3 units to the right instead?
3. How does a vertical translation compare to a horizontal translation in terms of graph movement?
4. What would be the equation if the graph was shifted 8 units to the right instead?
5. How does changing the coefficient of the absolute value affect the steepness of the graph?

Tip: Horizontal translations inside the absolute value function can be identified by examining the changes in the terms inside the absolute value.