## Precalculus (6th Edition)

RECALL: (1) The graph of the function $y=f(x) + k$ involves a vertical shift of $|k|$ units (upward when $k$ is positive, downward when $k$ is negative) of the parent function $y=f(x)$. (2) The graph of the function $y=f(x-h)$ involves a horizontal shift of $|h|$ units (to the right when $h$ is positive, to the left when $h$ is negative) of the parent function $y=f(x)$. (3) The graph of the function $y=-f(x)$ involves a reflection across the x-axis of the parent function $y=f(x)$. (4) The graph of the function $y=f(-x)$ involves a reflection across the y-axis of the parent function $y=f(x)$. (5) The graph of the function $y=a\cdot f(x)$ involves either a vertical stretch of the parent function $y=f(x)$ when $a \gt 1$ or a vertical shrink when $0 \lt a\lt 1$. (6) The graph of the function $y=f(ax)$ involves either a horizontal stretch of the parent function $y=f(x)$ when $0 \lt a \lt 1$ or a horizontal shrink when $a\gt 1$. The parent function of the given function is $y=x^2$. Graph this function. (Refer to the black graph below.) The given function can be written as $f(x) = -[x-(-1)]^2+3$. This equation is of the form $y=-f(x-h)+k$ Thus it involves the following graph transformations of the parent function $y=x^2$: (1) with $h=-1$, there is a 1-unit shift to the left (refer to the orange graph below); (2) a reflection across the x-axis (refer to the green graph below); and (3) with $k=3$, there is a 3-unit shift upward (refer to the blue graph in the answer part above.