1. **Problem statement:** We start with a function $y=f(x)$ and apply three transformations: - Vertical reflection about the x-axis - Horizontal stretch about the line $x=0$ by a factor of $\frac{1}{2}$ - Horizontal translation 5 units left 2. **Recall transformation rules:** - Vertical reflection about the x-axis changes $f(x)$ to $-f(x)$. - Horizontal stretch by a factor $k$ about $x=0$ changes $f(x)$ to $f(\frac{x}{k})$. - Horizontal translation 5 units left changes $f(x)$ to $f(x+5)$. 3. **Apply transformations step-by-step:** - Start with $f(x)$. - Horizontal stretch by $\frac{1}{2}$ means replace $x$ by $2x$ (since $k=\frac{1}{2}$, input changes as $x \to \frac{x}{k} = 2x$), so function becomes $f(2x)$. - Horizontal translation 5 units left means replace $x$ by $x+5$, so function becomes $f(2(x+5)) = f(2x+10)$. - Vertical reflection about the x-axis means multiply the whole function by $-1$, so function becomes $-f(2x+10)$. 4. **Final transformed function:** $$y = -f(2x + 10)$$ This equation represents the function after all three transformations applied in the order given.