Above, a function f is graphed in blue, and its derivative, f ′ is graphed with
a dashed line. Click and drag the graph of f around and see how the graph for f ′ is affected.
Finish the sentence: If the graph of f is shifted vertically by a units, then
the graph of f ′ is...
Let's make that idea mathematical. If a is any number, then f (x) + a looks
like f (x), just shifted vertically. Then
d
dx
(f (x) + a) =
d
dx
f (x) +
d
dx
a =
d
dx
f (x) = f ′(x)
Redo the previous two parts, writing something similar for horizontal shifts.
Vertical scaling
Above, a function f is graphed in blue, and its derivative, f ′ is graphed with
a dashed line. Observe how the graphs of f and f ′ are affected when you change the value of k.
Finish the sentence: If the f -graph is scaled vertically by a factor of k,
then the graph of f ′...
Express the above idea mathematically: If we know that
d
dx
f (x) = f ′(x), then
d
dx
(kf (x)) = ...
Horizontal scaling
Above, a function f is graphed in blue, and its derivative, f ′ is graphed with
a dashed line. Observe how the graphs of f and f ′ are affected when you change the value of k.
Horizontal scaling is tricky because the derivative is affected in two ways.
If the f -graph is scaled horizontally by a factor of k, what two things happen to
the graph of f ′?
One more tricky thing about horizontal scaling: if you want to scale f (x) horizontally
by a factor of k, then you make the function f (