Unit 3 Exponential, Logarithmic, Logistic Functions3.1 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 FunctionThe exponential function f with base b is defined by

f(x) = abx or y = abx

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) = abx or y = abx

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) = 2x g(x) = 10x h(x) = 3x+1

Base is 2 Base is 10 Base is 3

Determine formulas for the exponential functions g and h whose values are given in Table 3.2.

g(x) = abx

g(x) = 4(3)x

h(x) = abx

h(x) = 8(1/4)x

Characteristics of Exponential Functions

• The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.

• The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.

• If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing or growth function.

• If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing or decay function.

• f (x) = bx is a one-to-one function and has an inverse that is a function.

• The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis or y = 0 is a horizontal asymptote.

• The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.

• The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.

• If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing or growth function.

• If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing or decay function.

• f (x) = bx is a one-to-one function and has an inverse that is a function.

• The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis or y = 0 is a horizontal asymptote.

f (x) = bx

b > 1 f (x) = bx

0 < b < 1

Characteristics of Exponential Functionsf (x) = bx

b > 1 f (x) = bx

0 < b < 1

Describe each functions ending behavior using limits.

f (x) = bx

0 < b < 1f (x) = bx

b > 1

Transformations Involving Exponential Functions

• Shifts the graph of f (x) = bx upward c units if c > 0.

• Shifts the graph of f (x) = bx downward c units if c < 0.

g(x) = bx + cVertical translation

• Reflects the graph of f (x) = bx about the x-axis.

• Reflects the graph of f (x) = bx about the y-axis.

g(x) = -bx

g(x) = b-x

Reflecting

Multiplying y-coordintates of f (x) = bx by c,

• Stretches the graph of f (x) = bx if c > 1.

• Shrinks the graph of f (x) = bx if 0 < c < 1.

g(x) = c bxVertical stretching or shrinking

• Shifts the graph of f (x) = bx to the left c units if c > 0.

• Shifts the graph of f (x) = bx to the right c units if c < 0.

g(x) = bx+cHorizontal translation

DescriptionEquationTransformation

Complete Student CheckpointGraph: f (x) 2x

Use the graph of f(x) to obtain the graph of:

g(x) 2x 1

h(x) 2x 1

x y

-3 1/8-2 1/4

-1 1/20 1

1 22 4

3 8

f(x)

Horizontally shift 1 to the right

g(x)

Vertically shiftup 1

h(x)

The Natural Base eAn irrational number or natural base, symbolized by the letter e is approximately equal to 2.72; more accurately e = 2.718281828459…

f(x) = ex is called the natural exponential function2.71828...e -1

f (x) = ex

f (x) = 2x

f (x) = 3x

(0, 1)

(1, 2)

1

2

3

4

(1, e)

(1, 3)

Solve for x:

2x 22 3

2x 26

2x 43

x 6

Modeling San Jose’s populationThe 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) = abx use 1990 data as the initial value

g(x) = 782,248bx use 2000 data to calculate b, x is years after 1990 and g(x) is population.

894,943 = 782,248b10

894,943/782,248 = b10

894,943

782,24810 b

1.0135≈ b

San Jose’s population after 1990: g(x) = 782,248(1.0135)x

Modeling San Jose’s populationThe 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 wheny = 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: (18.313884, 1000000)

1990 + 18 = 2008

The population of San Jose will surpass 1 million persons in 2008.

Logistic Growth ModelThe 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.

f (x) c1 a bx

f (x) c1 a e kx

Modeling restricted growth of DallasBased on recent census data, the following is a logistic model for the population of Dallas, t years after 1900. According to this model, when was the population 1 million?

P(x) 1,301,614

1 21.603e 0.05055t

Use the graphing calculator to calculate the intersection wheny = 1,000,000

Intersection: (84.499296, 1000000)

1900 + 84 = 1984

The population of Dallas was 1 million in 1984.

Use graphing window [0, 120] by [-500000, 1500000]

Exponential and Logistic Functions