Vocabulary An exponential decay function has the form y = ab x, where a > 0 and 0 < b < 1. The base b of an exponential function is called the decay factor. The time required for a substance to fall to half its initial value is called half-life.
Example 1: Graph y = (1/2) x
Example 2: Graph y = -3(2/5) x
Example 3: Graph y = 3(1/2) x+1 – 2
Vocabulary Exponential Decay Model The amount y of the quantity after t years can be modeled by the equation y = a(1 – r) t, where a is the initial amount and r is the percent decrease expressed as a decimal. Note that the quantity 1 – r is the decay factor.
Example 4: The rate at which caffeine is eliminated from the bloodstream is about 15% per hour. An adult drinks a caffeinated soda, and the caffeine in his or her bloodstream reaches a peak level of 30 milligrams. Predict the amount, to the nearest tenth of a milligram, of caffeine remaining 1 hour after peak and 4 hours after the peak level.
Solution: In growth to obtain the growth factor we added our rate, so exponential decay we _________ the rate of decay from 100%. What is our decay factor? Write the expression for the caffeine level x hours after the peak level. 30(.85) x 85% or 0.85
Solution continued: What is the amount of caffeine remaining after 1 hour? 25.5 milligrams After 4 hours? 15.7 milligrams