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