What can you say about symmetry in this diagram?

first set of graphs

Now think about symmetry in this diagram.

second set of graphs

  • Is there a property relating to symmetry that is shared by all the graphs in each diagram?

  • Are there properties relating to symmetry that are only shared by some graphs?

We say that a function is even if its graph is symmetric about the \(y\)-axis. A function \(f(x)\) is even if and only if \(f(-x)=f(x)\) for each input value \(x.\)

We say that a function is odd if its graph has rotational symmetry about the origin, so that when you rotate the function through \(180^{\circ}\) about the point \((0,0)\) the graph looks the same. A function \(f(x)\) is odd if and only if \(f(-x)=-f(x)\) for each input value \(x.\)

For a function to be odd or even, its graph needs to have a particular type of symmetry, and about a particular line or point.

However, being odd or even is just one type of symmetry a function can have. There are plenty of functions whose graphs are symmetric in some way, but which are neither odd nor even. Do the graphs above suggest some examples?