## Derivatives and Graph Transformations

HELP

#### Translations

1. 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.
2. Finish the sentence: If the graph of `f` is shifted vertically by `a` units, then the graph of `f'` is...
3. 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)`]
4. Redo the previous two parts, writing something similar for horizontal shifts.

#### Vertical scaling

1. 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`.
2. Finish the sentence: If the `f`-graph is scaled vertically by a factor of `k`, then the graph of `f'`...
3. Express the above idea mathematically: If we know that `d/dx f(x) = f'(x)`, then `d/dx (k f(x)) = `...

#### Horizontal scaling

1. 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`.
2. 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'`?
3. 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) = `...?