Question

review: graph the following equations by hand. you must clearly graph or list two points on each graph! indicate the asymptote with a dashed line. then list the following information for each graph: domain (in interval notation), range (in interval notation), and the asymptote (va or ha) as an equation.
$y = \log_{3}(x - 2) + 4$
domain:
range:
asymptote:

Explanation:

Step1: Find Domain

For \( y = \log_3(x - 2)+4 \), the argument of the logarithm must be positive: \( x - 2>0 \), so \( x > 2 \). In interval notation, domain is \( (2, \infty) \).

Step2: Find Range

The range of a logarithmic function \( \log_b(u) \) is all real numbers, and vertical shifts don't change the range. So range is \( (-\infty, \infty) \).

Step3: Find Asymptote

For \( y=\log_b(x - h)+k \), the vertical asymptote is \( x = h \). Here, \( h = 2 \), so vertical asymptote (VA) is \( x = 2 \).

Step4: Find Points

• When \( x = 3 \): \( y=\log_3(3 - 2)+4=\log_3(1)+4 = 0 + 4=4 \), so point \( (3, 4) \).
• When \( x = 5 \): \( y=\log_3(5 - 2)+4=\log_3(3)+4 = 1 + 4=5 \), so point \( (5, 5) \).

Answer:

Domain: \( (2, \infty) \)
Range: \( (-\infty, \infty) \)
Asymptote: \( x = 2 \)
Points: \( (3, 4) \), \( (5, 5) \) (graph with VA \( x = 2 \), increasing, passing through these points)