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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