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 ={{1}\over{3}}#, #b=-3# and #c=-2#. Seeing as #a>0# the graph is a parabola that opens upward.

The intersection with the #y#-axis is equal to the value of the constant in the quadratic formula, which is equal to #-2#. That means that the coordinates of the intersection point with the #y#-axis are #\rv{0,-2}#.

The #x#-value of the vertex is given by #x=-\dfrac{b}{2 \cdot a}# and is equal to:
\[\begin{array}{rclrl}
x&=& -\dfrac{-3}{2 \cdot {{1}\over{3}}} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& {{9}\over{2}} &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x={{9}\over{2}}# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& {{1}\over{3}} \cdot {{9}\over{2}}^2 -3 \cdot {{9}\over{2}} -2
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{35}\over{4}} &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{{{9}\over{2}},-{{35}\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{4.5,-8.8}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
{{x^2}\over{3}}-3\cdot x-2 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{-3}-\sqrt{\left(-3\right)^2-4 \cdot {{1}\over{3}} \cdot -2}}{2 \cdot {{1}\over{3}}} &\vee& x=\dfrac{-{-3}+\sqrt{\left(-3\right)^2-4 \cdot {{1}\over{3}} \cdot -2}}{2 \cdot {{1}\over{3}}} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x={{9-\sqrt{105}}\over{2}} &\vee& x={{\sqrt{105}+9}\over{2}} \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{{{9-\sqrt{105}}\over{2}},0}# and #\rv{{{\sqrt{105}+9}\over{2}},0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{-0.6,0}# and #\rv{9.6,0}#.

The four points in the graph are: #\rv{0,-2}#, #\rv{{{9}\over{2}},-{{35}\over{4}}}#, #\rv{{{9-\sqrt{105}}\over{2}},0}# and #\rv{{{\sqrt{105}+9}\over{2}},0}#.