Use the graph of  f(x) = x to write an equation for the function represented by each graph.The x y-coordinate plane is given. A curve starts nearly vertically at the origin, goes down and right becoming less steep, passes through the point (4, −1⁄4), and exits the window in the fourth quadrant.


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.

Use the graph of f(x) = x to write an equation for the function represented by each graph. The x y-coordinate plane is given. A curve starts nearly vertically at the origin, goes down and right becoming less steep, passes through the point (4, −1⁄4), and exits the window in the fourth quadrant.

Discover how to transform the linear function f(x) = x by reflecting it over the x-axis and applying a vertical compression. This example illustrates the transformation process, leading to the equation f(x) = -1/16x. Suitable for students in Grades 8-10.

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