1. Applications of Exponentials Day 2

2. Determine increasing or decreasing, then combine with 100%
What is the general form for an exponential function?
  𝑦=𝑎∙ 𝑏 𝑥
What would we do if given a percent for our rate?
Determine increasing or decreasing, then combine with 100%

3. Growth and Decay from a Percent
Exponential Form increasing by a percent: 𝑦=𝑎∙(100+ ______ ) 𝑥
Example 1:
You have a loan worth $800 that has a yearly interest of 6%. Find the loan balance after 8 years.
𝑦=800(100+6% ) 𝑥 =800(106% ) 𝑥 =800(1.06 ) 𝑥
After 8 years: 800(1.06 ) 8 =

4. Growth and Decay from a Percent
Exponential Form decreasing by a percent: 𝑦=𝑎∙(100− ______ ) 𝑥  
Example 2:
The population of a city in 1998 was 100,000. The population has been decreasing by 2% every year ever since. Find the population after 6 years. What is the population after 11 years?
 𝑦=100000(100−2% ) 𝑥 =100000(98% ) 𝑥 =100000(.98 ) 𝑥
After 6 years: (.98 ) 6 =
After 11 years: (.98 ) 11 =

5. Rate of Increase or Decrease
From our total percentage, we can find how much the percent of change is. Remember we always start at 100%.
Example 3 Example 4 Example 5 Example 6
100%+2.5%
Rate of increase is 2.5%
100%-40%
Rate of decrease is 40%
200%
Take away 100%
Rate of increase is 100%
50%
Take away 100%
Rate of decrease is 50%
These are already broken up, so just convert to percent.
These are still combined, so convert to percent then subtract 100%

6. Finding Time from Total Amount
The last application of exponential functions is determining the time based on a total amount. The total amount represents y and your time is x.
 Example 7
You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by about 12%. How long until you have 10mg of caffeine?
𝑦=120(100−12% ) 𝑥 =120(88% ) 𝑥 =120(.88 ) 𝑥
Plug into your calculator to see when y=10.
About 19 hours.