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{5}}#, #b=-2# and #c=7#. 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 #7#. That means that the coordinates of the intersection point with the #y#-axis are #\rv{0,7}#.

The #x#-value of the vertex is given by #x=-\dfrac{b}{2 \cdot a}# and is equal to:
\[\begin{array}{rclrl}
x&=& -\dfrac{-2}{2 \cdot -{{1}\over{5}}} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -5 &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x=-5# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& -{{1}\over{5}} \cdot \left(-5\right)^2 -2 \cdot -5 +7
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& 12 &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{-5,12}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
-{{x^2}\over{5}}-2\cdot x+7 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{-2}-\sqrt{\left(-2\right)^2-4 \cdot -{{1}\over{5}} \cdot 7}}{2 \cdot -{{1}\over{5}}} &\vee& x=\dfrac{-{-2}+\sqrt{\left(-2\right)^2-4 \cdot -{{1}\over{5}} \cdot 7}}{2 \cdot -{{1}\over{5}}} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x=-2\cdot \sqrt{15}-5 &\vee& x=2\cdot \sqrt{15}-5 \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{-2\cdot \sqrt{15}-5,0}# and #\rv{2\cdot \sqrt{15}-5,0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{-12.7,0}# and #\rv{2.7,0}#.

The four points in the graph are: #\rv{0,7}#, #\rv{-5,12}#, #\rv{-2\cdot \sqrt{15}-5,0}# and #\rv{2\cdot \sqrt{15}-5,0}#.