The graph you described represents a transformation of the basic function $f(x) = x$. The transformation involves a reflection and a horizontal or vertical stretch or compression.

Step-by-Step Transformation:

1. Reflection over the x-axis: The graph starts by going downward, which suggests a reflection over the x-axis. This changes the function to $f(x) = -x$.

2. Vertical Compression: The curve is less steep, and it passes through the point $(4, -\frac{1}{4})$. This indicates that the function has been vertically compressed. A vertical compression by a factor of $k$ modifies the function to $f(x) = -kx$.

   To determine $k$, use the point $(4, -\frac{1}{4})$: $-k(4) = -\frac{1}{4}$ $k = \frac{1}{16}$

Thus, the equation of the function represented by the graph is:

$f(x) = -\frac{1}{16}x$

This function reflects the basic line $f(x) = x$ over the x-axis and applies a vertical compression by a factor of $\frac{1}{16}$.

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Would you like more details or have any questions? Here are five related questions to expand on this topic:

1. How do horizontal stretches and compressions affect the graph of a function?
2. What happens to the graph of $f(x) = x$ if you apply a vertical stretch by a factor of 2?
3. How does reflecting a function over the y-axis change its equation?
4. Can you explain the impact of shifting a graph horizontally by 3 units to the right?
5. How would you find the equation of a parabola given its vertex and a point on the graph?

Tip: When determining transformations, always pay close attention to how specific points on the graph relate to the original function to find the exact transformation factor.