Searching Intersections of

and ( is real number)

by algebraic approach

by Kaori Tabeta

In order to the observation, we notice the graph is always the central symmetry with the center point of the origin. We can observe or proof this feature according to search the intersections of and . By this observation, we also notice these equations haven't three intersections always. Through algebraic approach of the equation, proof the feature of the central symmetry and search the intersections of the equations.

Substance for

Then

When the equation is conclusion with .Therefore, if is any number, there is the intersection of the origin.

When then

When

because its discriminant is

so

Substance for then

Therefore, when and, intersections of the equations accord with.

When and,it is not true, then there is only intersection of the origin.

When that is to say

Then

and

So, when , the coordinates of the intersection of the equations without the origin are below.

Only the signs differ from coordinates of two intersections , so these two points are the central symmetry with the central point of the origin.

For these coordinate numbers are the real numbers :

So and

Or and

Because of and

For and ,when then ,when then

For and ,when then ,when then

Therefore, when , or

and when , or

As , then

So when , the intersection is the only (0,0)

Conclusion

As and or ,or as and or , then there are three intersections of

.

The first two points differ from only signs . It means central symmetry with center point of the origin.

As and , or as and , then there is an only intersection of the origin.