Answer
The required parabola is shown below.
Work Step by Step
Use the steps shown below to determine the graph of the quadratic equation.
Step 1:
The quadratic function can be written as $f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$ corresponding to which the graph is a parabola whose vertex is at $\left( h,k \right)$.
Thus, the vertex of the parabola $f\left( x \right)=-{{\left( x-1 \right)}^{2}}+4$ is $\left( h,k \right)=\left( 1,4 \right)$.
Step 2:
The parabola is also symmetric with respect to the line $x=h$. So, the provided parabola is symmetric to $x=1$.
Step 3:
If $a>0$ , the parabola opens upwards and if $a<0$ then the parabola opens downwards. Also, if $\left| a \right|$ is small, the parabola opens more flatly than if $\left| a \right|$ is large.
And, from the provided equation of the function, it is observed that the graph opens downwards as $a<0$.
Step 4:
So, we see that the above steps lead to the parabola that opens upwards and has vertex at $\left( 1,4 \right)$. Therefore, the required parabola is shown above.
