Symmetries of the gratification of a real function

Knowing the symmetry of a graph has a very important utility, since it helps us at the time of graphing, the explanation that this has is that by knowing the symmetry we do not need to know so many points to know the behavior that the graph will take.

Symmetry criteria in the graph of a real function

[1] The graph of an equation in x and y is symmetric about the y-axis if substituting x for -x in the equation yields an equivalent equation.

[2] The graph of an equation in x and y is symmetric about the x-axis if substituting y for -y in the equation results in an equivalent equation.

[3] The graph of an equation in x and y is symmetric about the origin if substituting x for -x and y for -y in the equation yields an equivalent equation.

There is an additional strategy to know if a graph is symmetric with respect to the y-axis, for example the graph of a polynomial is symmetric with respect to the y-axis if each of the terms of the polynomial has even exponent (or is a constant), for example the following polynomial:

The polynomial function shown is symmetric about the y-axis.

The counterpart is when a polynomial function has all its terms with odd exponent, that means that it is symmetric with respect to the origin, as for example:

Symmetry exercises

From exercises 19 to 26 of Larson's calculus book I am going to do exercise 27 of section P.I on page 8.

Find if the following function is symmetric with respect to the coordinate axes and the origin:

We test to see if there is symmetry with respect to the y-axis:

It must be satisfied that:

f(-x) = f(x)

Therefore:

We test to see if there is symmetry with respect to the x-axis:
It has to be fulfilled that:

-f(x) = f(x)

Therefore:

We test to see if there is symmetry with respect to the origin:

It must be fulfilled that:

-f(x)=f(-x) must be equal to f(x)

Therefore:

The conclusion of the case is that the function is only symmetric with respect to the y-axis, it has no symmetry with either the x-axis or the origin. We can corroborate this with the following graph in Geogebra:

Bibliographic Reference

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006