7-6 Exponential Functions

Quick Review

An exponential function involves repeated multiplication of an initial amount a by the same positive number b. The general form of an exponential function is y = a · b x , y equals eh middle dot , b to the x , comma  where a ≠ 0 , eh not equal to 0 comma   b > 0 , b greater than 0 comma  and b ≠ 1 . b not equal to 1 .

Example

What is the graph of y =   1 2 · 5 x ? y . equals , 1 half . middle dot , 5 to the x , question mark

Make a table of values. Graph the ordered pairs.

Exercises

Evaluate each function for the domain {1, 2, 3}.

49. f ( x ) = 4 x f open x close equals , 4 to the x
50. y = 0 . 01 x y equals 0 . , 01 to the x
51. y = 40 ( 1 2 ) x y equals 40 . open , 1 half , close to the x
52. f ( x ) = 3 · 2 x f open x close equals 3 middle dot , 2 to the x

Graph each function.

53. f ( x ) = 2 . 5 x f open x close equals 2 . , 5 to the x
54. y = 0.5 ( 0.5 ) x y equals 0.5 . open 0.5 close to the x
55. f ( x ) = 1 2 · 3 x f open x close equals , 1 half , middle dot , 3 to the x
56. y = 0 . 1 x y equals 0 . , 1 to the x

57. Biology A population of 50 bacteria in a laboratory culture doubles every 30 min. The function p ( x ) = 50 · 2 x p open x close equals 50 middle dot , 2 to the x  models the population, where x is the number of 30-min periods.

    1. How many bacteria will there be after 2 h?
    2. How many bacteria will there be after 1 day?

7-7 Exponential Growth and Decay

Quick Review

When a > 0 eh greater than 0  and b > 1 , b greater than 1 comma  the function y = a · b x y equals eh middle dot , b to the x  models exponential growth. The base b is called the growth factor. When a > 0 eh greater than 0  and 0 < b < 1 , 0 less than b less than 1 comma  the function y = a · b x y equals eh middle dot , b to the x  models exponential decay. In this case the base b is called the decay factor.

Example

The population of a city is 25,000 and decreases 1% each year. Predict the population after 6 yr.

y = 25 , 000 · 0 . 99 x Exponential decay function   = 25 , 000 · 0 . 99 6 Substitute  6  for x .   ≈ 23 , 537 Simplify. table with 3 rows and 3 columns , row1 column 1 , y , column 2 equals 25 comma 000 middle dot 0 . , 99 to the x , column 3 cap exponentialdecayfunction , row2 column 1 , , column 2 equals 25 comma 000 middle dot 0 . , 99 to the sixth , column 3 cap substitute . 6 , for , x . , row3 column 1 , , column 2 almost equal to 23 comma 537 , column 3 cap simplify. , end table

The population will be about 23,537 after 6 yr.

Exercises

Tell whether the function represents exponential growth or exponential decay. Identify the growth or decay factor.

58. y = 5.2 · 3 x y equals 5.2 middle dot , 3 to the x
59. f ( x ) = 7 · 0 . 32 x f open x close equals 7 middle dot 0 . , 32 to the x
60. y = 0.15 ( 3 2 ) x y equals , 0.15 . open , 3 halves , close to the x
61. g ( x ) = 1.3 ( 1 4 ) x g , open x close , equals 1.3 . open , 1 fourth , close to the x
62. Finance Suppose $2000 is deposited in an account paying 2.5% interest compounded quarterly. What will the account balance be after 12 yr?
63. Music A band performs a free concert in a local park. There are 200 people in the crowd at the start of the concert. The number of people in the crowd grows 15% every half hour. How many people are in the crowd after 3 h? Round to the nearest person.

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