Question

graph the equation by completing the table and plotting points. identify the domain.
$y = \ln x - 6$
part 1 of 3
complete the table. round to two decimal places when necessary.

x	y
0.5	-6.69
0.75	-6.29
1	-6.00
2	-5.31
3	-4.90
4	-4.61

alternate answer:

x	y
0.5	-6.69
0.75	-6.29
1	-6
2	-5.31
3	-4.9
4	-4.61

part: 1 / 3
part 2 of 3
graph the equation. plot the vertical asymptote and the points from the table.

Explanation:

Step1: Recall the parent function

The parent function is \( y = \ln x \), which has a vertical asymptote at \( x = 0 \) and domain \( (0, \infty) \). The function \( y=\ln x - 6 \) is a vertical shift (down 6 units) of the parent function \( y = \ln x \). Vertical shifts do not affect the vertical asymptote or the domain of logarithmic functions.

Step2: Determine the vertical asymptote

For \( y=\ln x - 6 \), the vertical asymptote remains \( x = 0 \) (since the argument of the logarithm is still \( x \), and the domain of \( \ln x \) is \( x>0 \), and shifting vertically doesn't change where the function is undefined).

Step3: Plot the points

We have the points from the table: \( (0.25, - 7.39) \), \( (0.5, - 6.69) \), \( (0.75, - 6.29) \), \( (1, - 6.00) \), \( (2, - 5.31) \), \( (3, - 4.90) \), \( (4, - 4.61) \). Plot these points on the coordinate plane. The graph of \( y=\ln x-6 \) will have the same shape as \( y = \ln x \) but shifted down 6 units, approaching the vertical asymptote \( x = 0 \) as \( x \) approaches 0 from the right, and increasing slowly as \( x \) increases.

Step4: Identify the domain

The domain of a logarithmic function \( y=\ln(u) \) is \( u>0 \). In the function \( y=\ln x - 6 \), the argument of the logarithm is \( x \), so we need \( x>0 \). Thus, the domain is all positive real numbers, or in interval notation, \( (0, \infty) \).

Answer:

To graph \( y = \ln x-6 \):

• Vertical Asymptote: Draw the vertical line \( x = 0 \) (dashed line, since it's an asymptote, not part of the graph).
• Plot Points: Plot the points from the table \( (0.25, - 7.39) \), \( (0.5, - 6.69) \), \( (0.75, - 6.29) \), \( (1, - 6.00) \), \( (2, - 5.31) \), \( (3, - 4.90) \), \( (4, - 4.61) \) and draw a smooth curve through them, approaching the vertical asymptote \( x = 0 \) as \( x \to 0^+ \) and increasing as \( x \) increases.
• Domain: The domain of \( y=\ln x - 6 \) is \( (0, \infty) \) (all real numbers \( x \) such that \( x>0 \)).