The red dots are the four dots from the question. These are calculated as follows:
The formula is already written in the form of #a \cdot x^2+b \cdot x +c# with #a =-5#, #b=5# and #c=9#. Seeing as #a<0# the graph is a parabola that opens downward.

The intersection with the #y#-axis is equal to the value of the constant in the quadratic formula, which is equal to #9#. That means that the coordinates of the intersection point with the #y#-axis are #\rv{0,9}#.

The #x#-value of the vertex is given by #x=-\dfrac{b}{2 \cdot a}# and is equal to:
\[\begin{array}{rclrl}
x&=& -\dfrac{5}{2 \cdot -5} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& {{1}\over{2}} &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x={{1}\over{2}}# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& -5 \cdot {{1}\over{2}}^2 +5 \cdot {{1}\over{2}} +9
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& {{41}\over{4}} &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{{{1}\over{2}},{{41}\over{4}}}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives: #\rv{0.5,10.2}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
-5\cdot x^2+5\cdot x+9 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{5}-\sqrt{5^2-4 \cdot -5 \cdot 9}}{2 \cdot -5} &\vee& x=\dfrac{-{5}+\sqrt{5^2-4 \cdot -5 \cdot 9}}{2 \cdot -5} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x={{5-\sqrt{205}}\over{10}} &\vee& x={{\sqrt{205}+5}\over{10}} \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{{{5-\sqrt{205}}\over{10}},0}# and #\rv{{{\sqrt{205}+5}\over{10}},0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{-0.9,0}# and #\rv{1.9,0}#.

The four points in the graph are: #\rv{0,9}#, #\rv{{{1}\over{2}},{{41}\over{4}}}#, #\rv{{{5-\sqrt{205}}\over{10}},0}# and #\rv{{{\sqrt{205}+5}\over{10}},0}#.