How do you graph - 2ln x?

To graph the function \( y = -2 \ln x \), follow these steps:

Understanding the Function

The function involves the natural logarithm, which is defined for \( x > 0 \). The negative sign and the coefficient of -2 will affect the shape and position of the graph.

Key Features of the Graph

• Domain: The function is defined for \( x > 0 \).
• Range: The output values can go from negative infinity to 0.
• Intercept: The graph crosses the y-axis at (1, 0) since \( \ln(1) = 0 \).

Steps to Graph

1. Start by plotting the point at \( x = 1 \), where \( y = 0 \).
2. Calculate additional points for values of \( x \) such as 0.5, 2, and 3:
   • For \( x = 0.5 \): \( y = -2 \ln(0.5) \approx 1.386 \)
   • For \( x = 2 \): \( y = -2 \ln(2) \approx -1.386 \)
   • For \( x = 3 \): \( y = -2 \ln(3) \approx -2.197 \)
3. Plot these points on a coordinate plane.
4. Draw a smooth curve through the points, noting that as \( x \) approaches 0, \( y \) approaches infinity, and as \( x \) increases, \( y \) decreases.

Final Touches

Label the axes and ensure the curve reflects the characteristics of the logarithmic function, showing a steep decline as \( x \) increases. This will give you a clear representation of \( y = -2 \ln x \).