The given function is $f(x) = 2x + 3$, where the slope is $2$ and the $y$-intercept is $3$. The problem asks how the function changes if the $y$-intercept becomes $-5$.

Analyzing the options:

1. A. "The graph of the line would be translated down 8 units."

   • This is correct because the $y$-intercept decreases from $3$ to $-5$, which is a change of $8$ units downward.

2. B. "The graph of the line would be translated up 8 units."

   • This is incorrect because the graph is moving downward, not upward.

3. C. "The slope of the line would change from increasing to decreasing."

   • This is incorrect because the slope does not change, it remains $2$, which is positive, indicating the line is still increasing.

4. D. "The slope of the line would change from decreasing to increasing."

   • This is incorrect as the slope remains unchanged at $2$.

5. E. "The function would become $f(x) = -5x + 3$."

   • This is incorrect because the slope would have to change to $-5$, but we are only changing the $y$-intercept.

6. F. "The function would become $f(x) = 2x - 5$."

   • This is correct because the slope stays $2$, and the new $y$-intercept is $-5$, which gives the equation $f(x) = 2x - 5$.

Correct answers:

• A and F.

Would you like a more detailed explanation on any part of this? Here are some related questions:

1. What is the effect of changing the slope in a linear function?
2. How can you calculate the change in the graph's position given a change in the $y$-intercept?
3. Why does a change in the $y$-intercept not affect the slope?
4. How can you tell if a line is increasing or decreasing from its slope?
5. How would changing both the slope and the intercept affect the graph of the function?

Tip: In a linear function, the slope controls the steepness and direction of the line, while the $y$-intercept determines where it crosses the vertical axis.