Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.
$$y = 4 + 2 x - x ^ { 2 }$$
Select the correct answer.

Question

Quizplus · Accepted Answer

We can start with the graph of $f(x) = -x^2$ and apply the transformations:
\begin{align*}
g(x) &= f(x-1) + 4 \
&= -(x-1)^2 + 4
\end{align*}
which corresponds to shifting the parabola $f(x)=-x^2$ to the right by 1 unit and up by 4 units. Then, we can apply the transformation:
\begin{align*}
y &= 2 x + g(x) \
&= 2x - (x-1)^2 + 4 \
&= -x^2 + 2x + 3
\end{align*}
which corresponds to a vertical shift up by 3 units. Thus, the final graph is a parabola opening downwards with vertex at $(1,3)$.

Option C is the only graph that fits this description.

Graph the Function by Hand, Not by Plotting Points, but by Starting