7.1 Notes – Modeling Exponential Growth and Decay

Checking off homework: 1) 6.7 Part 2 – 9, 11, 21-29(odd) 2) 6.8 1, 7-19(odd 3) 6.6-6.8 Test Review Bellwork: Make an x-y table for each function and draw as accurate a graph as you can for each function. 1) 𝑓 𝑥 = 2 𝑥 2) 𝑔 𝑥 = ( 1 2 ) 𝑥 3) How would you classify each function? Homework: Read 7.1. 7.1 Part 1 (2,3,7,8,11,13) (Label at least 3 points on each graph) 7.1 MORE Complete Solutions (in documents)

Homework: Read 7.1. 7.1 Part 1 (2,3,7,8,11,13) (Label at least 3 points on each graph) 7.1 MORE Complete Solutions (in documents)

2 −3 = 1 2 3 = 1 8 2 −2 = 1 2 2 = 1 4 2 −1 = 1 2 1 = 1 2 2 0 =1 2 1 =2 2 2 =4 2 3 =8

( 1 2 ) −3 = 2 3 =8 ( 1 2 ) −2 = 2 2 =4 ( 1 2 ) −1 = 2 1 =2 ( 1 2 ) 0 =1 ( 1 2 ) 1 = 1 2 ( 1 2 ) 2 = 1 2 2 2 = 1 4 ( 1 3 ) 2 = 1 2 3 2 = 1 8

Exponential Decay 𝑦=𝑎 𝑏 𝑥 𝑎=1 (starting value) 𝑏= 1 2 (b<1; decay factor) Exponential Growth 𝑦=𝑎 𝑏 𝑥 𝑎=1 (starting value) 𝑏=2 (b>1; growth factor)

What am I going to learn? Concept of an exponential function Models for exponential growth Models for exponential decay Meaning of an asymptote Finding the equation of an exponential function

Recall Independent variable is another name for domain or input, which is typically but not always represented using the variable, x. Dependent variable is another name for range or output, which is typically but not always represented using the variable, y.

What is an exponential function? Obviously, it must have something to do with an exponent! An exponential function is a function whose independent variable is an exponent.

What does an exponential function look like? y=ab Exponent and Independent Variable Dependent Variable Just some number that’s not 0 Why not 0? Base

The Basis of Bases The base of an exponential function carries much of the meaning of the function. The base determines if the function represents exponential growth or decay. The base is a positive number; however, it cannot be 1. We will return later to the reason behind this part of the definition .

Exponential Growth An exponential function models growth whenever its base > 1. (Why?) If the base b is larger than 1, then b is referred to as the growth factor.

What does Exponential Growth look like? Consider y = 2x Table of Values: Cool Fact: All exponential growth functions look like this! x 2x y -3 2-3 -2 2-2 ¼ -1 2-1 ½ 20 1 21 2 22 4 3 23 8 Graph:

Investigation: Tournament Play The NCAA holds an annual basketball tournament every March. The top 64 teams in Division I are invited to play each spring. When a team loses, it is out of the tournament. Work with a partner close by to you and answer the following questions.

Investigation: Tournament Play After round x Number of teams in tournament (y) 64 1 2 3 4 5 6 Fill in the following chart and then graph the results on a piece of graph paper. Then be prepared to interpret what is happening in the graph. Come up with a model for the data.

Investigation: Tournament Play After round x Number of teams in tournament (y) 64 1 32 2 16 3 8 4 5 6 Coming up with a model: 𝑦= 𝑎𝑏 𝑥 a=starting value = 64 b=growth(decay) factor =1/2 𝑦= 64( 1 2 ) 𝑥 What is wrong with this model?

𝒚= 𝟔𝟒( 𝟏 𝟐 ) 𝒙 What is wrong with this model? To improve the model, we just need to restrict the domain! Domain: 0≤𝑥≤6 AND 𝑥 must be a whole number. Defining variables is always a good idea too: Let 𝑥=𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑢𝑛𝑑𝑠 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒𝑑. and let 𝑦=𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑎𝑚𝑠 𝑙𝑒𝑓𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑜𝑢𝑟𝑛𝑒𝑦.

Exponential Decay An exponential function models decay whenever its 0 < base < 1. (Why?) If the base b is between 0 and 1, then b is referred to as the decay factor.

What does Exponential Decay look like? Consider y = (½)x Graph: Cool Fact: All exponential decay functions look like this! Table of Values: x (½)x y -2 ½-2 4 -1 ½-1 2 ½0 1 ½1 ½ ½2 ¼ 3 ½3 1/8

End Behavior Notice the end behavior of the first graph-exponential growth. Go back and look at your graph. as you move to the right, the graph goes up without bound. as you move to the left, the graph levels off-getting close to but not touching the x-axis (y = 0).

End Behavior Notice the end behavior of the second graph-exponential decay. Go back and look at your graph. as you move to the right, the graph levels off-getting close to but not touching the x-axis (y = 0). as you move to the left, the graph goes up without bound.

Asymptotes One side of each of the graphs appears to flatten out into a horizontal line. For exponential functions, an asymptote is a line that the graph approaches but never touches or intersects. More formal definition: In general, an asymptote is a line that a graph approaches as x or y increases in absolute value.

Asymptotes Notice that the left side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the curve will never take on a value of zero and will never be negative. y=0 is also called the x-axis.

Asymptotes Notice the right side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the graph will never take on a value of zero and will never be negative.

Let’s take a second look at the base of an exponential function Let’s take a second look at the base of an exponential function. (It can be helpful to think about the base as the object that is being multiplied by itself repeatedly.) Why can’t the base be negative? Why can’t the base be zero? Why can’t the base be one?

Examples Determine if the function represents exponential growth or decay. 1. 2. 3. Exponential Growth Exponential Decay Exponential Decay

Example 4 Writing an Exponential Function Write an exponential function for a graph that includes (0, 4) and (2, 1). (We’ll write out each step.)

Example 5 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). (Do each step on your own. We’ll show the solution step by step.) Use the general form. Substitute using (2, 2). Solve for a. Substitute using (3, 4). Substitute in for a.

Example 5 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). Simplify. Backsubstitute to get a. Plug in a and b into the general formula to get equation.

What’s coming up tomorrow? Applications of growth and decay functions using percent increase and decrease Translations of y = abx The number e Continuously Compounded Interest

Homework Problems 7.1 2,3,7,8,11,13 (Label at least 3 points on each graph)