Question

7-46. consider the graph at right. the parent function of the graph is ( y = log_2(x) ).
a. what is the equation of the vertical asymptote?
b. write a possible equation for this graph.
7-39. sketch a graph of the inequalities below and shade the solution region.
( y > |x + 3| )
( y leq 5 )

Explanation:

7-46a

Step1: Recall vertical asymptote of log function

The parent function \( y = \log_{2}(x) \) has a vertical asymptote at \( x = 0 \). For a transformed logarithmic function \( y=\log_{2}(x - h)+k \), the vertical asymptote is \( x = h \). From the graph, the vertical asymptote is at \( x = 2 \) (by observing the dashed line in the graph).

Step1: Analyze transformation of parent function

The parent function is \( y=\log_{2}(x) \). The vertical asymptote is \( x = 2 \), so there is a horizontal shift. The general form for a horizontal shift is \( y=\log_{2}(x - h) \), where \( h = 2 \) (since asymptote is \( x = 2 \)). Also, the graph passes through a point, let's assume it passes through \( (3,0) \) (from the graph). Plugging \( x = 3 \), \( y = 0 \) into \( y=\log_{2}(x - 2) \), we get \( 0=\log_{2}(3 - 2)=\log_{2}(1) \), which is true. So a possible equation is \( y=\log_{2}(x - 2) \).

Step1: Graph \( y > |x + 3| \)

The function \( y = |x + 3| \) is a V - shaped graph with vertex at \( (-3,0) \). Since \( y > |x + 3| \), we draw a dashed line for \( y = |x + 3| \) and shade the region above the V - shaped graph.

Step2: Graph \( y\leq5 \)

The line \( y = 5 \) is a horizontal line. Since \( y\leq5 \), we draw a solid line for \( y = 5 \) and shade the region below (including the line) this horizontal line.

Step3: Find the solution region

The solution region is the intersection of the two shaded regions: above \( y = |x + 3| \) (dashed line) and below or on \( y = 5 \) (solid line).

Answer:

The equation of the vertical asymptote is \( x = 2 \).

7-46b