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The problem in the image asks to find the equation of the resulting graph \( g(r) \) when the given functions are translated as specified. Let me explain the situation based on what is visible.

We have the following details:

1. **Part (a)**: The function \( f(x) = 2x + 3 \) is translated 4 units downwards.
2. **Part (b)**: The function \( f(r) = 3r - 4 \) is translated 2 units to the left.
3. **Part (c)**: The function \( f(r) = r^2 + 5r - 7 \) is translated 3 units upwards.
4. **Part (d)**: The function \( f(r) = 4r - 1 \) is translated 5 units to the right.

To translate these functions accordingly:

### (a) Translating \( f(x) = 2x + 3 \) 4 units downwards:
When a function is translated downwards by \( k \) units, we subtract \( k \) from the function.
\[ g(x) = f(x) - 4 = (2x + 3) - 4 = 2x - 1 \]

### (b) Translating \( f(r) = 3r - 4 \) 2 units to the left:
To translate a function \( k \) units to the left, we replace \( r \) with \( r + k \).
\[ g(r) = f(r + 2) = 3(r + 2) - 4 = 3r + 6 - 4 = 3r + 2 \]

### (c) Translating \( f(r) = r^2 + 5r - 7 \) 3 units upwards:
For an upward translation by \( k \) units, we add \( k \) to the function.
\[ g(r) = f(r) + 3 = (r^2 + 5r - 7) + 3 = r^2 + 5r - 4 \]

### (d) Translating \( f(r) = 4r - 1 \) 5 units to the right:
For a rightward translation by \( k \) units, we replace \( r \) with \( r - k \).
\[ g(r) = f(r - 5) = 4(r - 5) - 1 = 4r - 20 - 1 = 4r - 21 \]

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Would you like more details on any of these transformations, or do you have any questions?

Here are 5 related questions for further understanding:
1. How do horizontal translations affect the graph of a function?
2. How do vertical translations affect the graph of a quadratic function like \( f(x) = x^2 \)?
3. Can you explain how translations affect the intercepts of linear functions?
4. What other types of transformations can be applied to functions, besides translations?
5. How can translations be used to solve real-world problems involving graphs?

**Tip**: To easily remember translation rules, think of vertical shifts as changes to the \( y \)-axis and horizontal shifts as changes to the \( x \)-axis inputs!

Find the equation of the resulting graph g(r) for each of the given functions, after applying the specified translations.

Learn how to translate functions horizontally and vertically. This problem involves linear and quadratic functions, covering both left-right and up-down shifts.

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