Different equations can result in identical graphs. For example, [math]y-x=0[/math] and [math](y-x)(x^2+y^2)=0[/math] both have the same graph on the real xy-plane, a diagonal line through the origin. The first equation is a linear equation while the second is a cubic equation.

Edit: On the other hand, a function is determined by its values. If two functions have the same values, then they’re identical. That is, if functions [math]f[/math] and [math]g[/math] have the same domain, and for all [math]x[/math] in that domain it is the case that [math]f(x)=g(x)[/math], then [math]f[/math] and [math]g[/math] are identical functions even if they’re given by different formulas. So, for example, [math]\sin^2x[/math] and [math]1-\cos^2x[/math] describe the same function.