Question

a patient in a hospital is given an injection of medicine. the amount of medicine (in milligrams) left in the patients bloodstream after x hours can be modeled with an exponential function. the graph of this function is shown below. use the model to answer the parts to the right. (a) what is the y-intercept of this graph? the y-intercept represents the select. (b) for the first 20 hours, as time increases, the amount of medicine select. (c) give the equation of the asymptote. y = choose the statement that best describes the meaning of the asymptote. the patient can go home after 40 hours. the amount of medicine in the patients bloodstream does not fall below 0 mg. the amount of medicine in the patients bloodstream does not rise above 40 mg.

Explanation:

Step1: Find y-intercept

The y - intercept occurs at \( x = 0 \). From the graph, when \( x = 0 \) (time = 0 hours, just after injection), the y - value (amount of medicine) is 40 mg. The y - intercept represents the initial amount of medicine (at time \( x = 0 \), i.e., the amount of medicine immediately after the injection).

Step2: Analyze the trend for 20 hours

The graph is an exponential decay curve (since it is decreasing as \( x \) increases). So, for the first 20 hours, as time ( \( x \)) increases, the amount of medicine decreases.

Step3: Find the asymptote equation

The graph approaches the \( x \) - axis ( \( y = 0 \)) as \( x \) increases. So the equation of the asymptote is \( y = 0 \). For the meaning of the asymptote, since the amount of medicine can't be negative, the amount of medicine in the patient’s bloodstream does not fall below 0 mg.

Answer:

s:
(a) The y - intercept is 40. It represents the initial amount of medicine (immediately after injection).
(b) For the first 20 hours, as time increases, the amount of medicine decreases.
(c) The equation of the asymptote is \( y = 0 \). The correct statement is "The amount of medicine in the patient’s bloodstream does not fall below 0 mg."