Question

What value of k would create the function y=f(kx) on the graph if it is a transformed version of the function f(x)=x+2

Answer 1

To determine the value of \( k \) that would create the function \( y = f(kx) \) as a transformation of the function \( f(x) = x + 2 \), we need to understand what the transformation \( f(kx) \) does to the graph of the function.

The function \( f(x) = x + 2 \) is a linear function with a slope of 1 and a y-intercept of 2. The transformation \( f(kx) \) involves scaling the input \( x \) by a factor of \( k \). This kind of transformation can stretch or compress the graph horizontally depending on the value of \( k \):

- If \( |k| > 1 \), the graph will be compressed horizontally.
- If \( 0 < |k| < 1 \), the graph will be stretched horizontally.
- If \( k < 0 \), there will also be a reflection across the y-axis.

So \( f(kx) \) can be expressed as:

\[
f(kx) = kx + 2
\]

Now, if we want to specify a particular transformation of the graph, we need to consider what kind of effect we want:

1. If you're looking for a compression or stretch, you can choose any positive \( k \) (where \( k \neq 1 \)) to get a transformed version of the graph.
2. If you want to flip the graph over the y-axis, you can choose a negative value for \( k \).

Without a specific transformation goal provided, the value of \( k \) can be selected based on your desired outcome. For example:

- If you want a horizontal compression by a factor of 2, you would choose \( k = 2 \).
- If you want a horizontal stretch by a factor of 2, you would choose \( k = \frac{1}{2} \).
- If you want to reflect the function, you could choose \( k = -1 \).

Thus, the value of \( k \) can vary based on the specific transformation you want to achieve.