Question

What does the transformation f(x)↦4f(x) do to the graph of f(x)?

Answer 1

The transformation \( f(x) \mapsto 4f(x) \) scales the output values of the function \( f(x) \) by a factor of 4. This specifically affects the vertical position of the graph without altering its horizontal position.

Here's how this transformation changes the graph:

- **Vertical Stretch**: The graph of \( f(x) \) is stretched vertically by a factor of 4. If a point \( (x, y) \) lies on the graph of \( f(x) \), then after the transformation, the corresponding point on the graph of \( 4f(x) \) will be \( (x, 4y) \).

- **Effect on Points**: For example, if \( f(x) \) has a value of 2 at \( x = a \) (i.e., \( f(a) = 2 \)), then under the transformation \( 4f(x) \), the value at \( x = a \) will be \( 4 \times 2 = 8 \). Thus, the point \( (a, 2) \) on the original graph will move to the point \( (a, 8) \) on the transformed graph.

- **Overall Shape**: The overall shape of the graph remains the same, but it will be stretched further away from the x-axis by a factor of four. Peaks will be higher, and valleys will also be stretched downwards.

In summary, the transformation \( f(x) \mapsto 4f(x) \) stretches the graph of \( f(x) \) vertically by a factor of 4.