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Question

eliesaab · Answer

Let
$$
x=g(y)=a y^2 + b y + c
$$
We need to find a,b,c.
We need to find three equations in three unknowns (a,b,c) and solve them and we will be done

eliesaab · Answer

It seems that we only have two informations, but in fact we have three

eliesaab · Answer

The point (4,-2) is 
a point on the parabola and its vertex at the same time. (2 equations from here)
Also the point (3,4) is on the parabola. (one equation form here)
So we are all set

eliesaab · Answer

Interpreting the words into equations gives
$$

g'(-2)= b-4 a=0 \ 
g(-2)=4 a-2 b+c=4 \
g(4)= 16 a+4 b+c=3 \

$$

eliesaab · Answer

Solving them, you get
$$
\left(
\begin{array}{ccc}
 a=-\frac{1}{36} ,& b=-\frac{1}{9}, & c=\frac{35}{9} \
\end{array}
\right)
$$

eliesaab · Answer

Finally, the equation of the parabola is
$$
x=-\frac{y^2}{36}-\frac{y}{9}+\frac{35}{9}
$$

eliesaab · Answer

Are you with me?

eliesaab · Answer

attachment

eliesaab · Answer

The above is the graph. You can see the two points on it and one of them is the vertex.

eliesaab · Answer

You can also notice that your parabola intersects the y-axis at
y=-14 and y=10