Answer:
[tex]x=\dfrac34y^2[/tex]
Step-by-step explanation:
A graph with x-axis symmetry means that for every (x, y) point there is also a (x, -y) point on the graph, i.e. the x-axis acts like a mirror. Therefore, it is a sideways parabola.
The general form of this type of equation is [tex]x=ay^2+by+c[/tex]
Since we know that the curve has a vertex of (0, 0) then [tex]c=0[/tex]
Therefore, [tex]x=ay^2+by[/tex]
Given that (3, 2) is on the curve, then (3, -2) is also on the curve.
Substituting the given point (3, 2):
[tex]3=a(2)^2+b(2)[/tex]
[tex]3=4a+2b[/tex]
Substituting the given point (3, -2)
[tex]3=a(-2)^2+b(-2)[/tex]
[tex]3=4a-2b[/tex]
Adding the equations together to eliminate [tex]b[/tex]:
[tex]\implies 6=8a[/tex]
[tex]\implies 3=4a[/tex]
[tex]\implies a=\dfrac34[/tex]
Therefore,
[tex]x=\dfrac34y^2+by[/tex]
Again, substituting point (3, 2) means that b = 0
Therefore, the final equation is:
[tex]x=\dfrac34y^2[/tex]
