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 (k f(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(1/k x)`.
If we know that `d/dx f(x) = f'(x)`, then `d/dx f(1/k x) = `...?