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 =3#, #b=-4# and #c=-9#. 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 #-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:
\[\begin{array}{rclrl}
x&=& -\dfrac{-4}{2 \cdot 3} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& {{2}\over{3}} &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x={{2}\over{3}}# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& 3 \cdot {{2}\over{3}}^2 -4 \cdot {{2}\over{3}} -9
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{31}\over{3}} &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{{{2}\over{3}},-{{31}\over{3}}}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives: #\rv{0.7,-10.3}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
3\cdot x^2-4\cdot x-9 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{-4}-\sqrt{\left(-4\right)^2-4 \cdot 3 \cdot -9}}{2 \cdot 3} &\vee& x=\dfrac{-{-4}+\sqrt{\left(-4\right)^2-4 \cdot 3 \cdot -9}}{2 \cdot 3} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x={{2-\sqrt{31}}\over{3}} &\vee& x={{\sqrt{31}+2}\over{3}} \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{{{2-\sqrt{31}}\over{3}},0}# and #\rv{{{\sqrt{31}+2}\over{3}},0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{-1.2,0}# and #\rv{2.5,0}#.

The four points in the graph are: #\rv{0,-9}#, #\rv{{{2}\over{3}},-{{31}\over{3}}}#, #\rv{{{2-\sqrt{31}}\over{3}},0}# and #\rv{{{\sqrt{31}+2}\over{3}},0}#.