Why Circle is not a graph?

Question

My book says that though a circle is described by the equation x$^2$+y$^2$=a$^2$ where 'a' is the radius of the circle, But it is not a graph because a vertical line crosses the circle at more than 1 point. Why is this criteria of "vertical line crossing at more than one point" used as a qualifier for a graph and What does it signify?

Answer 1

I think your book is referring to the graph of a function. The vertical line test is typical to test this. A function can only have one output, y, for each unique input, x.

Answer 2

Why is a vertical line intersecting more than one point criterion?

A function, is a relation R such that if $(x,y_{1}) \in R$ and $(x,y_{2})$ in R then it must be that $y_{1}=y_{1}$

for example on a circle, you would have points say (1,5) but also (1,-5) lying on a circle, which would contradict the defintion