Exponential Equations and Expressions – Day 2

Key Concepts Another form of the exponential equation is , where a is the initial value, r is the rate of growth or decay, and t is the time. This formula is used when repeatedly finding increases or decreases based on a PERCENTAGE (%). In the formula above, use the + sign for growth and the – sign for decay. IMPORTANT! Notice that if a population grows by 2% then r is 0.02, but this is less than 1 and by itself does not indicate growth. We would have to add 1 to this to get our growth rate of 102% or 1.02.

Example 1 The population of a small town is increasing at a rate of 4% per year. If there are currently about 6,000 residents, about how many residents will there be in 5 years at this growth rate? First, make a list of what we know. Then, write your equation. Initial value= 6000 Time = 5 years Rate of increase = 4% = .04 There will be about 4892 residents.

Example 2 You want to reduce the size of a picture to place in a small frame. You aren’t sure what size to choose on the photocopier, so you decide to reduce the picture by 15% each time you scan it until you get it to the size you want. If the picture was 10 inches long to start, how long is it after 3 scans? First, make a list of what we know. Then, write your equation. Initial value= 10 Time = 3 scans Rate of decrease = 15% = .15 The 10 in picture frame is reduced to 6.14 inches

Example 3 Austin plans to open a savings account. He has $2000 but he is unsure of which savings account to use. In Account A, the interest rate will be 1.5% per year for 5 years. In Account B, the interest rate will be 2% per year for 3 years. Which account is the better option?   For A: What is the total savings if Austin chooses Account A? ___________________________________   For B: What is the total savings if Austin chooses Account B? ___________________________________   Which account has more money at the end of the term? ___________________________________

Sometimes, interest is compounded more often than once a year Sometimes, interest is compounded more often than once a year. Some common examples are when interested is compounded quarterly (4 times a year) or monthly (12 times a year). If this is the case, the formula may look slightly different  , where n is the only new variable representing the number of times compounded.

Example 4 An investment of $600 is compounded monthly at a rate of 4%. Use the formula to determine how much money will be in the account after 5 years. Determine your variables: a = 600 r = .04 n = 12 t = 5 Substitute these values into the equation and solve for y.

Key Concept - Graphing Graphing – We can graph these equations in the same way that we have graphed other exponential functions.  Let’s use the equation from Example 1: y = _______________ 1st – Create a table of values.  2nd – Complete the table of values by substituting values into the equation for x and solving for y. x y 6000 50 779.31 100 101.22 150 13.15

3rd – Plot these ordered pairs and connect them with a smooth curve 3rd – Plot these ordered pairs and connect them with a smooth curve. Be sure to label your graph! Total of residents 6000 5000 4000 3000 2000 1000 25 50 100 150 200 250 # of years

Activity – Population Change Copy on a sheet of paper :Due Tuesday On opposite sides of a major city two suburban towns are experiencing population changes. One town, Town A, is growing rapidly at 5% per year and has a current population of 39,000. Town B has a declining population at a rate of 2% per year. Its current population is 55,000. Economists predict that in 5 years the populations of these two towns will be about the same, but the residents of both towns are in disbelief. The economists also claim that ten years after that, Town A will double the size of Town B. Can you verify the predictions based on the data given? Do you think these predictions will come true?