To analyze the transformation from the function $f(x)$ to $h(x)$, let’s first examine both functions:

1. Given functions:

   • $f(x) = \frac{1}{4}x - 2$
   • $h(x) = -f(x)$

2. Step-by-Step Transformation Analysis:

   • Since $h(x) = -f(x)$, we are applying a vertical reflection of $f(x)$ across the x-axis.
   • To be explicit, substitute $f(x)$ into $h(x)$: $h(x) = -\left(\frac{1}{4}x - 2\right) = -\frac{1}{4}x + 2$
   • Therefore, the function $h(x) = -\frac{1}{4}x + 2$ represents a reflection of $f(x)$ across the x-axis.

3. Summary:

   • The transformation from the graph of $f(x)$ to $h(x)$ is a reflection across the x-axis. This changes the slope's sign and reflects any y-intercepts to the opposite side of the x-axis.

Would you like more details on the effects of this reflection or on the characteristics of the graph?

Here are some related questions for further exploration:

1. What would be the effect if we reflected $f(x)$ across the y-axis instead of the x-axis?
2. How would the transformation change if $h(x) = -f(x) + 3$ instead of just $h(x) = -f(x)$?
3. Can you describe how the slope and y-intercept change under a vertical reflection?
4. What would be the impact of adding a constant to $f(x)$ before reflecting it?
5. How would we transform $f(x)$ if we wanted to reflect it across both axes?

Tip: When reflecting a function vertically, only the y-values (outputs) of each point change signs, flipping the graph across the x-axis.