College Algebra 7th Edition

RECALL: (1) The function $y=f(x-h)$ involves a horizontal shift of $h$ units to the right of the parent function $f(x)$ when $h\gt0$. The function involves a horizontal shift of $|h|$ units to the left when $h \lt0$. (2) The function $y=-f(x)$ involves a reflection about the x-axis of the parent function $f(x)$. (3) The function $y=a\cdot f(x)$ involves a vertical stretch of the parent function $f(x)$ when $a\gt1$, and involves a vertical compression when $0 \lt a \lt 1$. (4) The function $y=f(x)+c$ involves a vertical shift of $c$ units upward of the parent function $f(x)$ when $c \gt 0$. The function will involve a vertical shift of $|c|$ units downward when $c \lt 0$. The parent function of the given function is $f(x)=\sqrt{x}$. The given function can be written $y=\frac{1}{2} \cdot f(x+4)-3$ where $f(x)$ is the parent function. Thus, the graph of the given function involves: (i) a horizontal shift of 4 units to the left, (ii) a vertical compression by a factor of $2$, and (iii) a vertical shift of 3 units downward of the parent function $f(x)=\sqrt{x}$. Therefore, to graph the given function: (1) Graph the parent function $y=\sqrt{x}$ . (Refer to the red graph in the attached image below) (2) Shift the parent function 4 units to the left. (Refer to the orange graph in the attached image below.) (3) Multiply $\frac{1}{2}$ to each y-values of the function in Step (2) while keeping the x-value. (Refer to the green graph in the attached image below.) (4) Shift the graph in Step (3) three units downward. (Refer to the blue graph in the attached image in the answer part above.)