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 analyze the statements about the function \( a(t) = 500(0.8)^t \):

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

   • This statement is false. The function is a decreasing exponential function, which means that \( a(t) \) decreases as \( t \) increases. Therefore, \( a \) does not increase at all.

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

   • This statement is true. To find the y-intercept, we evaluate the function at \( t = 0 \): \[ a(0) = 500(0.8)^0 = 500(1) = 500. \] So the y-intercept is indeed (0, 500).

3. Option #3: The range includes all real numbers.

   • This statement is false. The range of the function \( a(t) \) is from 0 to 500, not all real numbers, since the amount of medicine cannot be negative or exceed 500 mg.

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