Exploring Exponential Models. What you’ll learn To define exponential functions. Graph exponential functions. Solve applications of exponential growth and decay. Vocabulary Exponential function, exponential growth, exponential decay, asymptote, growth factor, decay factor

Take a note: An exponential function is a function with the general form . , In an exponential function the base b is a constant. The exponent x is the independent variable with domain the set of real numbers. Examples:

Problem 1:Graphing an Exponential Function. What is the graph Step 1: Make a table of values. Step 2: Plot and connect the points. 𝑥 𝑦 -4 -3 -2 -1 1 2 3

What is the graph of each function? Your turn: What is the graph of each function? Exponential Decay Exponential Growth The 𝑥−𝑎𝑥𝑖𝑠 is an asymptote -2 9 -1 3 1 0.3 -2 0.06 -1 0.25 1 4

Notes from the vocabulary There are two types of exponential behavior: exponential growth and exponential decay. For exponential growth, as the value of x increases, the value of y increases. For exponential decay, as the value of x increases, the value of y decreases, approaching zero. The exponential functions shown here are asymptotic to the axis. An asymptote is a line that a graph approaches as x increases or y increases in absolute value. Go back to the previous slide

Problem 2: Identifying Exponential Growth and Decay Take a note Problem 2: Identifying Exponential Growth and Decay Identify each function or situation as an example of exponential growth or decay. What is the y-intercept? a) 0<b<1 decay (0,12) Answers b) b>1 growth (0,0.25) c) 5% is the growth, y-intercept is 1000 is the initial investment

Your turn Identify each function or situation as an example of exponential growth or decay. What is the y –intercept? Answers Exponential growth;3 b) Exponential decay; 11 c) Exponential growth; 2000

Amount after t time periods A very important note: For the function If b>1 then is an exponential growth and b is the growth factor. A quantity that exhibits exponential growth increases by a constant percentage each period of time. That percentage increase r, written as a decimal is the rate of increase or growth rate.b=1+r If 0<b<1 then is an Exponential decay and b is the decay factor. The quantity decreases by a constant percentage each period of time. The percentage decrease r, is the rate of decay. Usually a rate of decay is expressed as a negative quantity, so b=r+1 Rate of growth Initial amount Amount after t time periods Number of time periods

Answer Problem 3:Modeling Exponential Growth. You invested $1000 in a savings account at the end of 6th grade. The account pays 5% annual interest. How much money will be in the account after six years? Answer Step 1: Determine if an exponential function is a reasonable model. The money grows at a fixed rate of 5% per year. An exponential model is appropriate. Step 2: Define the variables and determine the model. t= the number of years since the money was invested A(t)= the amount in the account each year. Step 3: Use the model to solve the problem

Your turn Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after 5 years? Answer:

Answer Problem 4: Using Exponential Growth: Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no additional deposits or withdrawals, how many years will it takes for the account to growth to at least $1500? Answer Step 1: Define the variables t= number of years and A(t)=the amount in the account after t years Step 2: Determine the model Step 3: Make a table Get the graphing calculator, made the table using the table feature. Find the input when the output is 1500. Answer: The acc. will no contain $1500 until the 9th year.

Your turn a)Suppose you invest in a savings account that pays 3.5% annual interest. When will the account contains at least $650. b) Use the table in Problem 4 to determine when that account will contain at least $1650. Answers b) After 11 years; the account contains $1710.34

Take a note: Exponential forms are often discrete (meaning discrete graph is just points, and the points are not connected in any way). Like for example problem 4, interest is paid only once a year so the graphs consist of individual points when t=1, t=2, t=3 and so on. To model a discrete situation using the exponential form you will need to find the growth or decay for factor b. So you will need to find the rate of change(remember you know the y-value for two consecutives x-values)

Problem 5: Writing an Exponential Form. The table shows the world population of Iberian Lynx in 2003 and 2004 Is this trend continues and the population is decreasing exponentially, how many Iberian lynx will be there in 2014? World population Of Iberian Lynx Year 2003 2004 Population 150 120

Step 1: Using the general form . Define the variables. x= the numbers of years since 2003 Y= the population of the Iberian Lynx. Step 2: Determine r Step 3: Use r to find b: Step 4: Write the model Step 5: Use the model to find the population in2014

Your turn: a)For the model in Problem 5, what will be The world population of Iberian lynx in 2020? b) If you graphed the model in Problem 5, will it ever cross the x-axis? Explain Answer: a) b) no, the function is asymptotic to the x-axis

CW/HW Form G