Warm-Up 1.3.2 Day 5This year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his hard work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents this situation. Solve it and show all work.

5001070 x40070 x7.5x

So school started more than 5 months ago.

Key Concepts• The general form of an exponential equation is

, where a is the initial value, b is the base, and x is the time. The final output value will be y.

• b must always be greater than zero.• If the base is greater than 1 (b > 1), then the

equation models exponential growth.• If the base is between 0 and 1 (0 < b < 1), then

the exponential equation represents exponential decay.

xbay

Additional Hints!

• Look for words such as double, triple, half, or quarter. These words give the number of the base.

• Look for words like initial or starting to substitute for a.

Example 1• A population of mice quadruples every year. If

a mouse nest started out with 2 mice, how many mice would there be after 4 years? Write an equation and then use it to solve the problem.

• First, make a list of what we know. • Then, write your equation.

Initial mice = 2So a = 2

Base = quad which means 4, so b= 4

X = every 6 months for 2 years so 4 times during that time

xbay 4)4)(2(y

512)256)(2( y

512 mice after 2 years

Example 2• In sporting tournaments, teams are eliminated

after they lose. The number of teams in the tournament then decreases by half with each round. If there are 16 teams left after 3 rounds, how many teams started out in the tournament?

• First, make a list of what we know.

• Then, write your equation.

Final number = 16Reduction = ½Time = 3

xbay 3

2

116

a

8

116 a a128

128 teams started out in the tournament.

Key Concept• Sometimes, values may increase after different

time periods. For instance, Example 1 tells us that the population of mice increases every ONE year. What if we were told that the population of mice quadrupled every 2 years?

• We can take this information and tweak our formula slightly to represent this time interval:

where 2 represents every 2 years. 2

x

bay

Example 3• The number of roses that bloom on your

rosebush doubles every 4 days. If the rose bush began with 5 blooms, how many roses would there be after 20 days?

• First, make a list of what we know.

• Then, write your equation.

Initial number= 5Doubles every 4daysTime period is 20 days

2

x

bay 4

20

)(5 by 4

20

)(5 by 160y

The rose bush will have 160 blooms after 20 days.