# Exponential Functions - An exponential pattern is something by malj

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```									                                           Exponential Functions

An exponential pattern is something that grows or decays by a multiple of some number. For
example, something that doubles, multiplies by 2 every time, and something that triples,
multiplies by 3 every time.

The general exponential equation is y = a(b)x, where a is the starting point (the number at
zero) and b is the number you are multiplying by. If b is bigger than 1, the equation is
growth, and if b is less than 1, the equation is decay.

Ex:      x        y                                      x         y
The starting point
0         2          The starting point          0     16
1        6                                      1     8
Multiplying by 3                                 Dividing by 2…so, multiplying by ½
2         18                                     2         4
3         54                                     3         2

This one’s growth                               This one’s decay
y = 2(3)x                                       y = 16(½)x

When dealing with percentage growth or decay, like interest problems or population
problems, the general formulas are:

Growth:         A = P(1 + r)t                   A = Final amount
P = Initial amount
Decay:          A = P(1 – r)t                   r = rate (in decimal form)
t = time in years

Ex:
1.     In 1800, Sallie Mae just found out that one of her ancestors invested \$300 in a
savings account that paid 4% interest annually. Find the account balance after the
year 1950.

P = \$300                        Equation A = 300 (1 + 0.04)150

r = 0.04                        using a calculator, typing in 300(1+0.04)150

t = 150                         the account balance is \$107676.80

2.     Suppose the acreage of forest is decreasing by 4% every year because of
development. If there are currently 6,000,000 acres of forest, determine the
amount of forest left after 15 years.

P = 6000000                     Equation A = 6000000(1 – 0.04)15

r = 0.04                        using a calculator, typing in 6000000(1 – 0.04)15

t = 15                          the forest acreage is 3,252,518.28 acres
Practice Problems

1.   y = 8(1.04)x

a) What is the initial amount?

b) What is the growth factor?

2.   y = 7(0.43)x

a) What is the initial amount?

b) What is the decay factor?

3.
x              y
1              1                 a) What is the initial amount?
2              4
b) What is the growth factor?
3             16
4            64                  c) What is the equation?
5            256

4.
x              y
1             81                 a) What is the initial amount?
2             27                 b) What is the decay factor?
3              9
4              3                 c) What is the equation?
5              1

Use the following tables to answer the questions #5 - 6.

A)      x           y           B)     x          y      C)     x        y     D)   x   y
1           1                 1         2              1        1          1   1
2           0.5                2         8              2        11         2   4
3        0.25                  3         18             3        21         3   16
4        0.125                 4         32             4        31         4   64

5. Which table represents an exponential growth function?

6. Which table represents an exponential decay function?

7. Write an equation that represents the data in the table.

x       1     2           3      4      5
y      75    30          12     4.8   1.92
8. Write an equation that represents the data in the table.

x         0        1      2       3     4
y       3.125    6.25    12.5    25     50

9.        For California, the population in 1900 was 1.77 million. Since then, the population has
grown at a rate of 3% per year. According to this rate, what was the population in
2000?

10.       A \$55,000 purchase depreciates at 16% each year. What would the value be after 10
years?

1. a) 8 b) 1.04

2. a) 7 b) 0.43

3. a) ¼ (remember it’s where x= 0)

b) 4

c) y = ¼ (4)x

4. a) 243

b) 1/3

c) y = 243 (1/3)x

5. D
6. A
7. y = 187.5(.4)x

8. y = 3.125(2)x

9. The population in 2000 was 34016978.61 people
10. The value is \$9619.57

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