Decay at a constant percentage rate can be described by 7-3 Exponential Decay Big Idea Decay at a constant percentage rate can be described by an expression of the form bg , where 0 < g < 1 and the variable is in the exponent. x Review the exponential growth, constant change, & exponential decay graphs and equations on p. 415.

In 1–3, graph the equation with a graphing calculator. Identify: Warm-Up In 1–3, graph the equation with a graphing calculator. Identify: a. the y-intercept of each graph. b. three points on the graph. c. the line that the graph approaches as x becomes very large. 1. y = 100(0.80) 2. y = 21,000(0.85) 3. y = 10(0.97) x x x 100 21,000 10 30 50 150 a. 100 a. 21,000 a. 10 (1, 80) (2, 64) (3, 51.2) (1, 17850) (2, 15172.5) (3, 12896.63) (1, 9.7) (2, 9.409) (3, 9.1267) c. x-axis c. x-axis c. x-axis

bg 120 (1-.15) 120 (.85) 53.244… 53 words x 5 5 Additional Example 1. Assume that each day after cramming, a student forgets 15% of the words known the day before. A student crams for a Spanish test on Thursday by learning 120 vocabulary words on Wednesday night. But the test is delayed from Thursday to Monday. If the student does not study more, how many words is he or she likely to remember on Monday? x bg 5 120 (1-.15) 5 120 (.85) 53.244… 53 words

2. In June 1953, the first Chevrolet Corvette rolled off the assembly line with a sticker price of approximately $3,000. Suppose its value depreciated by 9% each year after production. a. Find an equation that gives the car’s value y when it is x years old. b. What was the predicted value of the car in 1960? How close is this to the actual price of a 1953 Corvette, which was $1,640, in 1960? c. Graph the car’s value for the interval 0 ≤ x ≤ 7. d. What was the predicted value of the car in 2006? How close is this to the actual price of a 1953 Corvette, which was $59,900 in 2006? x a. y = 3000 · (.91) c. Next page 7 53 b. y = 3000 · (.91) d. y = 3000 · (.91) y = $1550.28 y = $20.24 1640 – 1550.28 = $89.72 59900 – 20.24 = $59879.76

y 3100 3000 2900 2800 2700 2600 2500 2400 2300 2200 2100 2000 1900 1800 1700 1600 1500 c.  x y 3000 1 2730 2 2484 3 2261 4 2057 5 1872 6 1704 7 1550 Value ($)   Z x 0 1 2 3 4 5 6 7 8 9 10 Age of Car

3. Caffeine is a drug found in a wide variety of food products consumed by Americans. In fact, more than half of all adult Americans consume at least 300 milligrams of caffeine every day. Once in the body, it takes about 6 hours for half of the caffeine to be eliminated. Suppose the pattern of eliminating caffeine from the body continues after it is ingested at one time; that is, half of the remaining caffeine is eliminated every 6 hours. a. Write an equation to describe y, the amount of caffeine in the bloodstream after x six-hour periods have passed. b. Make a calculator table for the equation from Part a. Use the table to find when approximately 1 milligram of caffeine remains in the body. c. How much caffeine remains after 108 hours? d. According to the equation, when will the amount of caffeine in the body be zero? b. x y 1 2 3 4 5 6 300 150 75 37.5 18.75 9.375 4.6875 x a. y = 300 · .5 c. (x = 108 ÷ 6 = 18) 18 y = 300 · .5 y = .001 mg x = 0 d. Never