Section 3.8 Going Viral: Exponential Functions
Questions • What is an exponential function? • What is the equation of an exponential function? • How can we write the equation of an exponential function from information given?
Definition A quantity that grows (or decays) by a constant percent change every fixed amount of input is representable by an exponential function.
Definition Such a relationship has formula y = abxwhere a is the initial value of the quantity (when x = 0) and b, which must be positive and not equal to 1, is called the growth or decay factor (or the multiplier).
Definition Such a relationship has formula y = abxwhere a is the initial value of the quantity (when x = 0) and b, which must be positive and not equal to 1, is called the growth or decay factor(or the multiplier). If the quantity grows by 100r% for every 1 unit of input, then b = 1 + r. If the quantity decays by 100r% for every 1 unit of input, then b = 1 –r.
Exercise 1 At the beginning of a certain experiment, the number of bacteria is 500. An antibacterial agent is introduced, and then the number of bacteria decreases by 12.25% each hour. Write a formula for the number of bacteria remaining after x hours.
Exercise 1 At the beginning of a certain experiment, the number of bacteria is 500. An antibacterial agent is introduced, and then the number of bacteria decreases by 12.25% each hour. Write a formula for the number of bacteria, B, remaining after x hours.
Exercise 2 In 2007, Facebook had $153 million in revenue, and for the next decade its revenue grew by about 34.84% annually. Give a formula for the revenue, R, x years after 2007.
Exercise 2 In 2007, Facebook had $153 million in revenue, and or the next decade its revenue grew by about 34.84% annually. Give a formula for the revenue, R, x years after 2007.
Graphs of abx The graph of an exponential function passes through the point (0, a). The graph of an exponential function has the x-axis as a horizontal asymptote. Whether the graph is increasing or decreasing depends on the value of b.
Graphs of abx The graph of an exponential function passes through the point (0, a) and has one of two forms: 0 < b < 1 b > 1
Formula From 2 Points Suppose we know two points (x1, y1) & (x2, y2) are on an exponential function. • . • Substituting b and one of the ordered pairs into the equation y = abx, we can then solve for a.
For Example Find the exponential function passing through the points (1, 6) and (4, 48). Thus,
Exercise 3 Two hours after a certain computer virus was launched in the office, 26 computers in the company were infected. Four hours later, 184 of the computers were infected. Assuming the number of computers infected is growing exponentially, find a formula for N, the number infected x hours after the initial launch.
Exercise 3 Two hours after a certain computer virus was launched in the office, 26 computers in the company were infected. Four hours later, 184 of the computers were infected. Assuming the number of computers infected is growing exponentially, find a formula for N, the number infected x hours after the initial launch.
