You'll notice that every quadratic equation has an equal sign, while quadratic functions don't have any (e.g. x+2x-3 function, 2x=-x+3 equation)
quadratic functions also have equal sign, that is, f(x) = ax^ + bx + c. :-)
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A quadratic equation is a second-degree polynomial (largest power is 2), usually written in standard form, ax² + bx + c = 0.  

A quadratic function is a relation between two variables wherein for every value of the independent variable (usually "x", the domain), there is exactly one corresponding value for the dependent variable (usually y, the range).  Function is given for the purpose of graphing. 

Quadratic funtion  is expressed as:
f(x) = ax² + bx + c

f(x) is actually y.  So it's also  y = ax² + bx + c