Explain why you cannot use the axes of symmetry to distinguish between the quadratic functions

That is because there are many quadratic functions that have the same axis of symmetry.  If a function is multiplied by a constant factor the axis of symmetry remains the same but the curved part of  the graph widens or narrows.

For example  x^2 + x - 6  and 2x^2 + 2x - 12  The last function will have same axis of symmetry and same roots as the first but the curved part will be narrower.

Also if we multiply the first function by -1 the curve will flip about the x axis but will have the same axis of symmetry