# Vertical Shifts and Derivatives

## Vertical Shifts and Derivatives

In the app above enter a formula for a function f(x) in the input box. It will be graphed in blue. Manipulate the value of C by its slider or input box. The graph of g(x) = f(x) + C will appear in green. Note that the graph of g is just a vertical shift of the graph of f(x) by C. Manipulate the value of a by its slider or input box. The tangent lines to the two functions for x = a are graphed. Their slopes are also computed and displayed. What do you notice? Since these two graphs are just vertical shifts of each other the tangent lines will also just be vertical shifts of each other. This makes them parallel lines. Parallel lines have the same slope. Therefore, g'(a) = f '(a). This works regardless of the value of a so g'(x) = f'(x). Check on the check box for g'(x) = f '(x) to see this common derivative function graphed in red. Check on the Proof checkbox to see the algebraic proof of this property. This property is particularly important when we think of reversing the process of finding a derivative, i.e. finding an antiderivative. Notice that the process of finding a derivative is a function itself. This function takes a function as an input and returns its one and only derivative function. However, the process of differentiation is not a one-to-one function. There are infinitely many different functions that produce the same derivative. All of these antiderivatives are vertical shifts of each other. For example, the derivative of f(x) =x^2+C is f'(x) = 2x. So the family of antiderivatives of h(x) = 2x is k(x) = x^2 +C for any real constant C. Taking antiderivatives gives not one antiderivative, but rather an entire family of antiderivatives all of which differ by a constant term, i.e. they differ by a vertical shift.