First, let's look at the geometric transformation that occurs when we multiply a function by a constant c (equivalently replace [i]y[/i] by [i]y/c[/i] in a formula). [br][br]In the App[br] Enter a formula of your choice in the input box for [i]f[/i]([i]x[/i]). It is graphed in blue if its checkbox is checked.[br] Manipulate the value of [i]c[/i] via its slider or input box.[br] Check the checkbox for [i]cf[/i]([i]x[/i]) to see this transformed function graphed in green.[br][br]Look at the values for the preimage point ([i]a[/i],[i]f[/i]([i]a[/i])) and its image ([i]a[/i], [i]cf[/i]([i]a[/i])). Notice that they have the same [i]x[/i]-coordinate, but the[i] y[/i]-coordinate of the image is[i] c[/i] times the [i]y[/i]-coordinate of the preimage. [br]Check the checkbox for Illustrate Strain to see vectors representing the displacement from the [i]x[/i]-axis for [i]f [/i]and cf a[i]t x = a[/i]. Notice that the displacement vector for [i]cf[/i] is simply [i]c [/i]times the displacement vector for [i]f[/i]. The technical term for such a geometric transformation is a vertical strain.[br][br]The result is that the graph of [i]f[/i]([i]x[/i]) has been stretched (|[i]c[/i]|> 1) or compressed (|[i]c[/i]| < 1) by a factor of |[i]c|[/i] vertically from the [i]x[/i]-axis to obtain the graph of [i]g[/i]([i]x[/i]) = [i]cf[/i]([i]x[/i]). If [i]c[/i] is negative, then the graph of the original[br]function is also reflected over the [i]x[/i]-axis to find the graph of the image function. [br]

Look at the values of f '(a) and cf '(a) given in the app. What is the relationship between these values? Make a conjecture and test it out with different values of [i]a[/i] and [i]c[/i].

Check the Constant Multiple Derivative Rule checkbox to see the rule you should have discovered.[br][br]Now check the Tangent Lines checkbox to see the tangent lines to [i]f [/i]and [i]cf [/i]drawn when [i]x = a[/i]. Notice that the slopes of these lines are the signed lengths of the vertical displacement vectors, showing how much the tangent line goes up or down when we go to the right one unit. Notice that the vertical displacement vector for the tangent line to c[i]f[/i] is [i]c[/i] times the vertical displacement vector for the tangent line to [i]f[/i]. Therefore, the slope of the tangent line to [i]cf[/i] is [i]c[/i] times the slope of the tangent line to [i]f[/i].

Check the checkbox for Proof to see a formal algebraic proof. The only things that are used in the proof are the definition of cf, the definition of the derivative, the distributive property (to factor), and the constant multiple rule for limits.