Why do we need the constant of integration?

Say that we have a differentiated equation f'(x), and we want to find the orginal equation f(x) from this. We would have to use integration, as differentiation and integration are the reverse of each other. For example: f'(x) = x² -5x +7   <---- the dash in f'(x) just means this is the derivative of a function f(x) We want to find f(x), so we have to integrate f'(x): f(x) = x³/3 - 5x²/2 + 7x    <---- I haven't shown how to integrate here as it isn't the question, however I can do this too! Our f(x) currently only tells us what the shape of the curve is, it does not tell us the position of the curve, (does it intersect the axes?). Therefore, our f(x) is not entirely correct: we need to add the constant of integration: f(x) = x³/3 - 5x²/2 + 7x + c An extra point: To find out the value of c, we need the coordinates of a point on the curve. Say our f(x)=y: y = x³/3 - 5x²/2 + 7x + c If we have the coordinates of a point on the curve (x,y), we can substitute them into our equation to find out the value of c and hence determine exactly what the curve looks like.