https://file.aiquickdraw.com/mathbot/file/1727200824882fjpqhrru.jpg

To solve this, we need to determine the equation of the line shown in the graph and identify which of the given options represents a line that is **perpendicular** to it.

### Step 1: Equation of the Line in the Graph
From the graph, the line crosses the y-axis at \( y = 1 \) (the y-intercept). Additionally, the line seems to pass through the point \( (-5, 0) \) (the x-intercept). Using these two points, we can calculate the slope \( m \) of the line:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{0 - (-5)} = \frac{1}{5}
\]

So, the equation of the line in the graph is:
\[
y = \frac{1}{5}x + 1
\]

### Step 2: Slope of Perpendicular Line
For two lines to be perpendicular, the product of their slopes must equal \( -1 \). Therefore, if the slope of the line in the graph is \( \frac{1}{5} \), the slope of a line perpendicular to it would be the **negative reciprocal** of \( \frac{1}{5} \), which is \( -5 \).

### Step 3: Identifying the Correct Equation
Now, let's check the given equations and their slopes:

1. \( y = \frac{1}{4}x - 2 \) → slope = \( \frac{1}{4} \)
2. \( y = 5x + 4 \) → slope = \( 5 \)
3. \( y = -\frac{1}{4}x - 1 \) → slope = \( -\frac{1}{4} \)
4. \( y = -5x - 2 \) → slope = \( -5 \)

The equation \( y = -5x - 2 \) has a slope of \( -5 \), which is perpendicular to the slope of the line in the graph.

### Conclusion
The correct answer is:
\[
\boxed{y = -5x - 2}
\]

Would you like further details or have any questions? 

Here are some related questions to deepen your understanding:
1. How do you derive the equation of a line given two points?
2. What is the geometric interpretation of slope?
3. Can two lines be parallel if they have different y-intercepts?
4. How do you find the slope of a line from its equation?
5. What are the conditions for two lines to be perpendicular?

**Tip:** Remember, when dealing with perpendicular lines, always look for the negative reciprocal of the given slope.

Which of these equations represents a line that is perpendicular to the line in the graph below?

Learn how to determine the equation of a line that is perpendicular to another line on a graph. This example uses slope calculations and applies the perpendicular slope property to find the correct answer.

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