Unit 3 Exponential, Logarithmic, Logistic Functions 3.1 Exponential and Logistic Functions (3.1) The exponential function f (x) = 13.49(0.967) x – 1 describes the number of O-rings expected to fail, f (x), when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of O-rings expected to fail at this temperature. Because the temperature was 31°F, substitute 31 for x and evaluate the function at 31. f (x) = 13.49(0.967) x – 1 f (31) = 13.49(0.967) 31 – 1 f (31) =3.77 About 4 of the O-rings are expected to fail at this temperature.

Definition of the Exponential Function The exponential function f with base b is defined by f(x) = ab x or y = ab x where a is the nonzero initial value of f (the value at x = 0), b is a positive constant other than 1 and x is any real number. The exponential function f with base b is defined by f(x) = ab x or y = ab x where a is the nonzero initial value of f (the value at x = 0), b is a positive constant other than 1 and x is any real number. Here are some examples of exponential functions. f (x) = 2 x g(x) = 10 x h(x) = 3 x+1 Base is 2 Base is 10Base is 3

Determine formulas for the exponential functions g and h whose values are given in Table 3.2. g(x) = ab x g(x) = 4(3) x h(x) = ab x h(x) = 8(1/4) x

Characteristics of Exponential Functions The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing or growth function. If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing or decay function. f (x) = b x is a one-to-one function and has an inverse that is a function. The graph of f (x) = b x approaches but does not cross the x-axis. The x-axis or y = 0 is a horizontal asymptote. The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing or growth function. If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing or decay function. f (x) = b x is a one-to-one function and has an inverse that is a function. The graph of f (x) = b x approaches but does not cross the x-axis. The x-axis or y = 0 is a horizontal asymptote. f (x) = b x b > 1 f (x) = b x 0 < b < 1

Characteristics of Exponential Functions f (x) = b x b > 1 f (x) = b x 0 < b < 1 Describe each functions ending behavior using limits. f (x) = b x 0 < b < 1 f (x) = b x b > 1

Transformations Involving Exponential Functions Shifts the graph of f (x) = b x upward c units if c > 0. Shifts the graph of f (x) = b x downward c units if c < 0. g(x) = b x + c Vertical translation Reflects the graph of f (x) = b x about the x-axis. Reflects the graph of f (x) = b x about the y-axis. g(x) = -b x g(x) = b -x Reflecting Multiplying y-coordintates of f (x) = b x by c, Stretches the graph of f (x) = b x if c > 1. Shrinks the graph of f (x) = b x if 0 < c < 1. g(x) = c b x Vertical stretching or shrinking Shifts the graph of f (x) = b x to the left c units if c > 0. Shifts the graph of f (x) = b x to the right c units if c < 0. g(x) = b x+c Horizontal translation DescriptionEquation Transformation

Complete Student Checkpoint Graph:Use the graph of f(x) to obtain the graph of: x y -31/8 -21/4 1/ f(x) Horizontally shift 1 to the right g(x) Vertically shift up 1 h(x)

The Natural Base e An irrational number or natural base, symbolized by the letter e is approximately equal to 2.72; more accurately e = … f(x) = e x is called the natural exponential function f (x) = e x f (x) = 2 x f (x) = 3 x (0, 1) (1, 2) (1, e) (1, 3)

Solve for x:

Modeling San Jose’s population The population for San Jose in 1990 was 782,248 and in 2000 it was 894,943. Assuming growth is exponential, when will the population of San Jose surpass 1 million persons? g(x) = ab x use 1990 data as the initial value g(x) = 782,248b x use 2000 data to calculate b, x is years after 1990 and g(x) is population. 894,943 = 782,248b ,943/782,248 = b ≈ b San Jose’s population after 1990: g(x) = 782,248(1.0135) x

Modeling San Jose’s population The population for San Jose in 1990 was 782,248 and in 2000 it was 894,943. Assuming growth is exponential, when will the population of San Jose surpass 1 million persons? Use the graphing calculator to calculate the intersection when y = 1,000,000 San Jose’s population after 1990: g(x) = 782,248(1.0135) x On the calculator use CALC, then INTERSECT and follow the directions. Intersection: ( , ) = 2008 The population of San Jose will surpass 1 million persons in 2008.

Logistic Growth Model The mathematical model for limited logistic growth is given by or Where a, b, c and k are positive constants, with and 0 < b < 1, c is the limit to growth. From population growth to the spread of an epidemic, nothing on Earth can grow exponentially indefinitely. This model is used for restricted growth.

Modeling restricted growth of Dallas Based on recent census data, the following is a logistic model for the population of Dallas, t years after According to this model, when was the population 1 million? Use the graphing calculator to calculate the intersection when y = 1,000,000 Intersection: ( , ) = 1984 The population of Dallas was 1 million in Use graphing window [0, 120] by [ , ]

Exponential and Logistic Functions