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Do there exist graphs of equations which are symmetric with respect to the $y$ -axis and origin, and if there are, do they exhibit any other kinds of symmetry?

Yes, also symmetry with respect to the $x$ -axis.

Algebra

Chapter 1

Functions and their Applications

Section 7

More on Functions

Functions

Campbell University

McMaster University

Harvey Mudd College

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Is there a function whose …

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Do the graphs of all strai…

their graphs, which, uh, symmetric with respect to both the Y axis and the our region. A circle is such an example. You can check it very easily if a graph is symmetric with respect to both wide access and the origin, we can conclude that this graph is also symmetric with respect to the X axis. Now let's prove this statement. If a graph is symmetric with respect to the Y axis, if they replace X rays, minor sex, then the Ukrainian of this graph kids and changed the photograph is symmetric with respect to their region. We can replace X with minus X and Y rays manners y at the same time. And this real keep the equation of the graph entrance. So f we assume that f X y is the equation of this graph. We first use a condition that this graph is symmetric with respect to the Y axis. Then we get f X y equals F minus X y. Next, we use the condition that this graph is symmetric with respect to the origin we replace, minus the eggs with X and y with minus what and this gives f X y equals F X manners. Wine knows that this is a condition that the graph is symmetric with respect to the X axis and we finished our proof here.

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