Question 1: Calculate the equation of the vertical and horizontal asymptotes for the function y = 5 / (x - 2).

Answer: {"vertical":"x=2","horizontal":"y=0"}

Question 2: Plot the graph of y = 3 / (x + 1). Identify its asymptotes.

Answer: {"vertical":"x=-1","horizontal":"y=0"}

Question 3: A company's production cost per unit is modeled by C(x) = 100 / (x - 50). Find the cost asymptote and interpret its meaning in a real-world context.

Answer: {"asymptote":"x=50","interpretation":"As the number of units approaches 50, the cost per unit increases dramatically, indicating a minimum production threshold."}

Question 4: Determine the asymptotes of the function y = (2x + 3) / (x - 4).

Answer: {"vertical":"x=4","horizontal":"y=2"}

Question 5: Explain why the graph of y = 1 / (x^2 - 9) has vertical asymptotes, and identify their locations.

Answer: {"locations":"x=3 and x=-3","explanation":"Vertical asymptotes occur where the denominator equals zero, so at x^2 - 9=0, which gives x=±3."}

Question 6: A tank leaks at a rate inversely proportional to the water level. The rate is modeled by R(h) = k / h, with k=20. Find the asymptote of the rate as the water level h becomes very large.

Answer: {"asymptote":"R(h) approaches 0 as h approaches infinity"}

Question 7: Construct a graph on the grid illustrating y = 4 / (x - 1) and label its asymptotes.

Answer: {"vertical":"x=1","horizontal":"y=0"}

Question 8: A scientist models the reaction rate by R = 10 / (t - 5). What is the significance of the vertical asymptote at t=5?

Answer: {"interpretation":"At t=5, the reaction rate becomes infinite, indicating a critical point or impossibility of the process at that time."}

Question 9: Find the asymptotes of y = (x^2 + 2x + 1) / (x - 2).

Answer: {"vertical":"x=2","horizontal":"y=x+3"}

Question 10: Identify and explain any possible mistake if a student claims y = 1 / (x^2 - 4) has only one vertical asymptote at x=2.

Answer: {"correction":"The function has vertical asymptotes at x=2 and x=-2 because the denominator x^2 - 4 factors as (x-2)(x+2)."}