The correct answer is D. y=tan(x+2)−77.
To determine the correct answer, let's analyze each option and its corresponding graph
A. y=tan(x−7)+2: This function represents a vertical shift of 2 units upward of the standard tangent function, with a horizontal shift of 7 units to the right. Its graph would have vertical asymptotes at x=7+πn, where n is any integer.
B. y=tan(x−7)−2: This function represents a vertical shift of 2 units downward of the standard tangent function, with a horizontal shift of 7 units to the right. Its graph would have vertical asymptotes at x=7+πn, where n is any integer.
C. y=tan(2(x+7))−2: This function represents a horizontal compression by a factor of 1/2, a horizontal shift of 7 units to the left, and a vertical shift of 2 units downward. Its graph would have vertical asymptotes at x=−7+πn/2, where n is any even integer.
D. y=tan(x+2)−77: This function represents a vertical shift of 77 units downward of the standard tangent function, with a horizontal shift of 2 units to the left. Its graph would have vertical asymptotes at x=−2+πn, where n is any integer.
Comparing the given graph with the descriptions of each option, we can see that the graph matches the description of D. y=tan(x+2)−77. The graph has vertical asymptotes at x=-2+πn, consistent with the horizontal shift of 2 units to the left, and the graph is shifted 77 units downward.
Therefore, the correct answer is D. y=tan(x+2)−77