[Mathematics] Symmetric with respect to axis

There are 3 major type of symmetric about an axis:

1. symmetric with respect to the x-axis
2. symmetric with respect to the y-axis
3. symmetric with respect to the the origin

The advantage to know an equation is symmetric is you can use the result to predict behavior on the other side.

In addition to using graph to find out the function is symmetric with respect to axis, we can also use check it symmetric alone axis by replacing points.

Let use this function as example:

y²=x-1

Examine symmetric about y-axis
We can replace (x,y) with (-x,y) to examine symmetric about y-axis
By replacing x to -x, we get y²=(-x)-1 from y=x-1

y²=-x-1

Since y²=-x-1 is not same to y²=x-1, the graph would not symmetric about y-axis.


Examine symmetric about x-axis
We can replace (x,y) with (x,-y) to examine symmetric about x-axis
By replacing y to -y, we get -y²=x-1 from y²=x-1

(-y)²=x-1
 (y)²=x-1

Since (-y)²=x-1 is same to y²=x-1, the graph is symmetric about x-axis.


Examine symmetric about origin
We can replace (x,y) with (-x,-y) to examine symmetric about origin
By replacing y to -y, we get (-y)²=(-x)-1 from y²=x-1

(-y)²=(-x)-1
    y²=-x-1

Since (-y)²=(-x)-1 is not same to y²=x-1, the graph is not symmetric about origin.

Finally, let take a peek about how y2=x-1 look like in graph: