1. **Problem Statement:** Identify the transformations of the function $g(x) = -|x + 3| + 4$ from the parent function $f(x) = |x|$. 2. **Parent Function:** The parent function is $f(x) = |x|$, which is a V-shaped graph with vertex at the origin $(0,0)$. 3. **Transformation Rules:** - Horizontal shifts: $|x - h|$ shifts the graph $h$ units right if $h > 0$, left if $h < 0$. - Vertical shifts: $f(x) + k$ shifts the graph $k$ units up if $k > 0$, down if $k < 0$. - Reflection: A negative sign outside the absolute value reflects the graph over the x-axis. 4. **Apply transformations to $g(x)$:** - Inside the absolute value, $x + 3$ means a horizontal shift left by 3 units. - The negative sign outside reflects the graph vertically (over the x-axis). - The $+4$ outside shifts the graph up by 4 units. 5. **Domain and Range:** - Domain of $g(x)$ is all real numbers: $$(-\infty, \infty)$$ because absolute value functions are defined everywhere. - To find the range, note the vertex is at $(-3, 4)$ and the graph opens downward due to reflection. - The maximum value is 4, so the range is $$(-\infty, 4]$$. 6. **Summary:** - Transformations: shift left 3, reflect vertically, shift up 4. - Domain: $$(-\infty, \infty)$$ - Range: $$(-\infty, 4]$$