Use the image to answer the question.

Question

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.

Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?

Option #1: As t increases, a increases slowly at first and then quickly.

Option #2: The y-intercept of the function is (0,500)
.

Option #3: The range includes all real numbers.

(1 point)
Option #__ is the true statement.

Answer 1

To evaluate the statements about the function \( a(t) = 500(0.8)^t \), we can go through each option:

Option #1: As \( t \) increases, \( a \) increases slowly at first and then quickly.
This is incorrect because the function is decreasing exponentially, not increasing. As \( t \) increases, \( a(t) \) decreases.

Option #2: The y-intercept of the function is \( (0,500) \).
This is correct. When \( t = 0 \), \( a(0) = 500(0.8)^0 = 500 \). Therefore, the y-intercept is indeed \( (0,500) \).

Option #3: The range includes all real numbers.
This is incorrect. The range of the function is \( (0, 500] \), meaning it includes values greater than 0 and up to 500 but does not include all real numbers.

Thus, the true statement is:

Option #2.