Answer
Refer to the blue graph below.
Work Step by Step
The given function can be written as:
$f(x) = -\frac{1}{2}(x-1)^2+3$
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 $y=x^2$.
The given function can be written $y=-\frac{1}{2}f(x-1)+3$ where $f(x)$ is the parent function.
Thus, the graph of the given function involves:
(i) a horizontal shift of $1$ units to the right,
(ii) a reflection about the x-axis,
(iii) a vertical compression, and
(iv) a shift of $3$ units upward
of the parent function $f(x)=x^2$.
Therefore, to graph the given function:
(1) Graph the parent function $y=x^2$ .
(Refer to the red graph in the attached image below)
(2) Shift the graph of the parent function 1 unit to the right.
(Refer to the orange graph in the attached image below.)
(3) Reflect the graph in Step (2) 3 about the x-axis.
(Refer to the green graph in the attached image below)
(4) Multiply $\frac{1}{2}$ to each y-value of the graph in Step (3) while retaining the x-coordinate of each point.
(Refer to the black graph in the attached image below.)
(5) Shift the graph in Step (4) 3 units upward.
(Refer to the blue graph in the attached image in the answer part above.)
