8.1 Exponential Growth

A Review of Linear Growth In Algebra 1, almost every real life situation that you encountered was linear. This means that to make mathematical predictions about the situation, you used a linear equation such as y = mx + b. Linear Growth: Amount of change (Slope) Initial Amount (Starting Point) Linear Growth assumes that the amount of change during each period is the same. This is rarely true in real-life.

A Review of Linear Growth Example (from Algebra 1): An airplane is flying at an altitude of 5000 feet. As it prepares to land, it descends 25 feet per second. 1.) What is an equation that models the altitude of the airplane? 2.) What is the altitude of the plane after 30 seconds? Linear Growth: Initial Amount (Starting Point) Amount of change (Slope)

Exponential Growth Exponential growth is similar to linear growth except exponential growth grows by multiplying each period rather than adding. Exponential Growth: Initial Amount (Starting Point) Percent of change (Growth factor)

Graphs of Exponential Growth Here is a graph of a simple exponential growth equation. The graph gets flatter and flatter as x gets more negative Initial Amount (Starting Point) Every period the amount triples. The graph gets steeper and steeper as x gets bigger. When x = 0, y = 2.

Graphs of Exponential Growth Here is a graph of a simple exponential growth equation. The graph gets fatter and flatter as x gets bigger The graph gets steeper and steeper as x gets more negative. Initial Amount (Starting Point) Every period the amount gets cut in half. When x = 0, y = 8.

Growth Factors What does the graph of the following function look like? Why does the graph stay flat? Why doesn’t it go up or down?

Growth Factors What does the graph of the following function look like? Why does the graph stay flat? Why doesn’t it go up or down? What if the growth factor is greater than one? This is the graph of: y = 3 (1.1)x If the growth factor is greater than one, the graph goes up. This is called exponential growth.

Growth Factors What does the graph of the following function look like? Why does the graph stay flat? Why doesn’t it go up or down? What if the growth factor is less than one? This is the graph of: y = 3(0.9)x This is the graph of: y = 3 (1.1)x If the growth factor is less than one, the graph goes down. This is called exponential decay.

Growth Factors Growth Decay When the growth factor is greater than one, the equation represents growth. The amount of growth is equal to the difference between the growth factor and one. Decay When the growth factor is less than one, the equation represents decay. The amount of decay is equal to the difference between the growth factor and one.

What is the growth factor? 10% decay Growth factor = 1.10 Growth factor = 0.90 25% growth 25% decay Growth factor = 1.25 Growth factor = 0.75 20.9% growth 20.9% decay Growth factor = 1.209 Growth factor = 0.791

How much growth or decay is represented by this growth factor?

What is the growth factor? 350% decay Growth factor = 2 Decay cannot exceed 100%. Why? 525% growth Growth factor = 6.25 Because you cannot lose more than you start with. 237.85% growth Growth factor = 3.3785

How much growth or decay is represented by this growth factor?

An ant colony had ten ants on March 1st An ant colony had ten ants on March 1st. Its population has been growing 65% every week. What equation models the population of ants as it grows from week to week? By the end of April, eight weeks have gone by, how many ants are now in the colony? 549.378 (549 ants)

Some doctors are testing the effectiveness of a new antibiotic Some doctors are testing the effectiveness of a new antibiotic. At 2:00 PM, they place the antibiotic in a Petri dish with 18 grams of bacteria. The antibiotic kills the bacteria at a rate of 12% per hour. What equation models the amount of bacteria from hour to hour? At 7:00 PM, how much bacteria remains in the dish? 9.499 grams

Finding an exponential equation that passes through points. Find the exponential equation that passes through: (0,12) & (3,1.5) Pick an easy point to fill in for x & y. b0 = 1, so a = 12 Plug in the other point and solve for b.

Finding an exponential equation that passes through points. Find the exponential equation that passes through: (0,12) & (3,1.5) Plug in the other point and solve for b. So the equation is:

Finding an exponential equation that passes through points. Find the exponential equation that passes through: (2,20) & (6,320) In this case, plugging in doesn’t eliminate either a or b. Plug in both separately and then stack them. Divide to eliminate a.

Finding an exponential equation that passes through points. Find the exponential equation that passes through: (2,20) & (6,320) Simplify these fractions Solve for b.

Finding an exponential equation that passes through points. Find the exponential equation that passes through: (2,20) & (6,320) Plug in b to solve for a. So the equation is:

Find the equation of the exponential that passes through (0,10) & (2,16)

Find the equation of the exponential that passes through (1,9) & (2,8)

Find the equation of the exponential that passes through (2,12) & (6,192)

Find the equation of the exponential that passes through (5,7) & (11,12)