Question

1. can a graph have more than one symmetry? if so, sketch an example of a graph (not necessarily of a function) that has more than one type of symmetry.

Explanation:

Step1: Recall symmetry types

Symmetry types include symmetry about the x - axis, y - axis, and origin.

Step2: Consider an example

The equation of a circle centered at the origin $x^{2}+y^{2}=r^{2}$ (where $r>0$) has multiple symmetries.

• Symmetry about the x - axis: Replace $y$ with $-y$ in the equation $x^{2}+y^{2}=r^{2}$, we get $x^{2}+(-y)^{2}=r^{2}$, which simplifies to $x^{2}+y^{2}=r^{2}$, the original equation. So it is symmetric about the x - axis.
• Symmetry about the y - axis: Replace $x$ with $-x$ in the equation $x^{2}+y^{2}=r^{2}$, we get $(-x)^{2}+y^{2}=r^{2}$, which simplifies to $x^{2}+y^{2}=r^{2}$, the original equation. So it is symmetric about the y - axis.
• Symmetry about the origin: Replace $x$ with $-x$ and $y$ with $-y$ in the equation $x^{2}+y^{2}=r^{2}$, we get $(-x)^{2}+(-y)^{2}=r^{2}$, which simplifies to $x^{2}+y^{2}=r^{2}$, the original equation. So it is symmetric about the origin.

Step3: Sketch the circle

To sketch the circle $x^{2}+y^{2}=r^{2}$ (for example, if $r = 2$), mark points on the x - axis at $(2,0)$ and $(-2,0)$ and on the y - axis at $(0,2)$ and $(0, - 2)$. Then draw a smooth curve passing through these points to form a circle centered at the origin $(0,0)$.

Answer:

Yes, a graph can have more than one symmetry. For example, the graph of the circle $x^{2}+y^{2}=r^{2}$ is symmetric about the x - axis, y - axis and the origin.