To determine what transformation converts the graph of $f(x) = -5x - 5$ into the graph of $g(x) = -x - 1$, we need to compare the two equations and look for changes in the structure.

Step 1: Compare the forms of the two functions

1. The function $f(x) = -5x - 5$ is in slope-intercept form:
   $f(x) = mx + b$, where:

   • $m = -5$ (slope),
   • $b = -5$ (y-intercept).

2. The function $g(x) = -x - 1$ is also in slope-intercept form:
   $g(x) = mx + b$, where:

   • $m = -1$ (slope),
   • $b = -1$ (y-intercept).

Step 2: Identify the transformations

• Change in slope: The slope of $f(x)$ is $-5$, and the slope of $g(x)$ is $-1$. The slope has changed from $-5$ to $-1$, which means the graph has undergone a vertical stretch (by a factor of 5) followed by a vertical compression (by a factor of 5), or it could be interpreted as a simple change in steepness.

• Change in y-intercept: The y-intercept of $f(x)$ is $-5$, and the y-intercept of $g(x)$ is $-1$. This suggests a vertical translation of the graph upward by $4$ units (because $-1 - (-5) = 4$).

Step 3: Conclusion

The transformation from $f(x) = -5x - 5$ to $g(x) = -x - 1$ can be described as:

1. A vertical compression by a factor of 5 (changing the slope from $-5$ to $-1$).
2. A vertical translation upward by 4 units (changing the y-intercept from $-5$ to $-1$).

Would you like more details on how these transformations work? Or have any further questions?

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Here are 5 related questions to explore:

1. How does a vertical stretch/compression affect the slope of a line?
2. What is the general effect of a vertical translation on the graph of a function?
3. How do you identify transformations when given two functions?
4. Can you describe the effect of a horizontal shift on the graph of a linear function?
5. How do reflections in the x-axis or y-axis alter the equation of a line?

Tip: When comparing linear functions, always start by comparing the slope and y-intercept to identify the main transformations.