Here we illustrate how a smooth function (such as sin(x)) can be approximated by a quadratic function near a point A lying on the graph of the function.
By dragging A, you can see that the quadratic function depends on A. Choose other functions by entering them in the box. Try other trigonometric functions, polynomials, functions involving square roots, exponentials and logarithms ... What happens if the function itself is a quadratic function? Check out Taylor polynomials and Taylor's theorem on the web ...