1. **Stating the problem:** We want to find two different combinations of transformations that change the function $f(x) = 5x + 3$ into $g(x) = 15x - 12$. 2. **Understanding transformations:** Transformations can include vertical/horizontal shifts, stretches/compressions, and reflections. 3. **Check the relationship between $f(x)$ and $g(x)$:** $$g(x) = 15x - 12 = 3(5x) - 12 = 3f(x) - 21$$ This means $g(x)$ is a vertical stretch of $f(x)$ by a factor of 3, then shifted down by 21 units. 4. **First combination:** - Vertical stretch by a factor of 3: $f(x) \to 3f(x) = 15x + 9$ - Vertical shift down by 21: $3f(x) - 21 = 15x + 9 - 21 = 15x - 12$ 5. **Second combination:** We can also express $g(x)$ as $f(3x) - 15$ because: $$f(3x) = 5(3x) + 3 = 15x + 3$$ Then shifting down by 15: $$f(3x) - 15 = 15x + 3 - 15 = 15x - 12$$ This corresponds to: - Horizontal compression by a factor of $\frac{1}{3}$ (replace $x$ by $3x$) - Vertical shift down by 15 6. **Summary:** - Combination 1: Vertical stretch by 3, then vertical shift down by 21 - Combination 2: Horizontal compression by $\frac{1}{3}$, then vertical shift down by 15 7. **Matching with options:** Option A matches these two combinations exactly. **Final answer:** Option A