Graph Transformation A93Ad2

1. **State the problem:** We need to find the correct transformation rule for a graph that undergoes the following changes: - Shifted 8 units to the right - Stretched vertically by a factor of 2 - Reflected in the x-axis - Shifted 3 units upward 2. **Recall the transformation rules:** - Horizontal shift right by $h$ units: replace $x$ by $x - h$ inside the function, so $f(x) \to f(x - h)$. - Vertical stretch by a factor of $a$: multiply the function by $a$, so $f(x) \to a f(x)$. - Reflection in the x-axis: multiply the function by $-1$, so $f(x) \to -f(x)$. - Vertical shift up by $k$ units: add $k$ outside the function, so $f(x) \to f(x) + k$. 3. **Apply each transformation step-by-step:** - Start with $f(x)$. - Shift right 8 units: $f(x) \to f(x - 8)$. - Stretch vertically by 2: $f(x - 8) \to 2 f(x - 8)$. - Reflect in x-axis: $2 f(x - 8) \to -2 f(x - 8)$. - Shift up 3 units: $-2 f(x - 8) \to -2 f(x - 8) + 3$. 4. **Final transformation rule:** $$g(x) = -2 f(x - 8) + 3$$ 5. **Compare with the options:** - Option a: $2 f(x - 8) + 3$ (no reflection, so incorrect) - Option b: $-2 f(x - 8) - 3$ (shift down 3, incorrect) - Option c: $-2 f(x + 8) + 3$ (shift left 8, incorrect) - Option d: $-2 f(x - 8) + 3$ (matches our result) - Option e: None of the answers (incorrect) **Answer:** Option d is correct. **Final answer:** $\boxed{g(x) = -2 f(x - 8) + 3}$