Reflection Vertical Stretch 924786

1. **State the problem:** We start with the function $f(x) = |x|$ and want to find a function $g(x)$ that represents a reflection of $f(x)$ in the x-axis and a vertical stretch by a factor of 4. 2. **Recall the transformations:** - Reflection in the x-axis changes $f(x)$ to $-f(x)$. - Vertical stretch by a factor of 4 changes $f(x)$ to $4f(x)$. 3. **Apply the transformations step-by-step:** - Reflect $f(x)$ in the x-axis: $-f(x) = -|x|$. - Stretch vertically by 4: $4 \times (-|x|) = -4|x|$. 4. **Write the final function:** $$ g(x) = -4|x| $$ 5. **Explanation:** The original graph $f(x) = |x|$ is a V-shaped graph opening upwards. Reflecting it in the x-axis flips it upside down, so it opens downwards. Then stretching it vertically by 4 makes it 4 times taller (or deeper) than the original. The resulting graph $g(x) = -4|x|$ is a V-shaped graph opening downwards, centered at the origin, and vertically stretched by 4 times.