Warm Up Warm Up Plug in points (of your choosing) and sketch the graph of each function on the same coordinate plane. Use different colors if possible. a f (x) = 2 x b g(x) = 4 x c h(x) = 2 x d j(x) = 4 x Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 1 / 17
Exponential Functions and Their Graphs Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 2 / 17
Exponential Functions and Their Graphs 1 Exponential Functions Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 3 / 17
Exponential Functions and Their Graphs 1 Exponential Functions 2 Graphs of Exponential Functions Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 3 / 17
Exponential Functions and Their Graphs 1 Exponential Functions 2 Graphs of Exponential Functions 3 The Natural Base e Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 3 / 17
Exponential Functions and Their Graphs 1 Exponential Functions 2 Graphs of Exponential Functions 3 The Natural Base e 4 Application Compound Interest Radioactive Decay Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 3 / 17
Definition of Exponential Function Definition of Exponential Function The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 4 / 17
Definition of Exponential Function Definition of Exponential Function The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number. The base a = 1 is excluded because it yields f (x) = 1 x = 1. This is actually a constant function, since it is always equal to 1, so it is not an exponential function. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 4 / 17
Graphs of y = a x and y = a x Practice Plug in points (of your choosing) and sketch the graph of each function on the same coordinate plane. Use different colors if possible. a f (x) = 2 x b g(x) = 4 x c h(x) = 2 x d j(x) = 4 x Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 5 / 17
Characteristics of Exponential Functions Investigation Consider the graphs of f (x) = 2 x and h(x) = 2 x from earlier. a For each function, determine the domain, range, intercepts, and asymptotes. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 6 / 17
Characteristics of Exponential Functions Investigation Consider the graphs of f (x) = 2 x and h(x) = 2 x from earlier. a For each function, determine the domain, range, intercepts, and asymptotes. b For each function, determine when the function is increasing and when the function is decreasing. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 6 / 17
Characteristics of Exponential Functions Investigation Consider the graphs of f (x) = 2 x and h(x) = 2 x from earlier. a For each function, determine the domain, range, intercepts, and asymptotes. b For each function, determine when the function is increasing and when the function is decreasing. c For each function, determine if the function is continuous. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 6 / 17
One-to-One Property From the previous investigation, we can determine that the graph of an exponential function is either always increasing or always decreasing. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 7 / 17
One-to-One Property From the previous investigation, we can determine that the graph of an exponential function is either always increasing or always decreasing. As a result, the graph passes the Horizontal Line Test, meaning that the functions are always one-to-one. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 7 / 17
One-to-One Property From the previous investigation, we can determine that the graph of an exponential function is either always increasing or always decreasing. As a result, the graph passes the Horizontal Line Test, meaning that the functions are always one-to-one. One-to-One Property For a > 0 and a 1, a x = a y if and only if x = y. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 7 / 17
Using the One-to-One Property Example Solve the equation 9 = 3 x+1 Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 8 / 17
Using the One-to-One Property Practice Solve the equation ( 1 2) x = 8 Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 9 / 17
Transformations of Graphs of Exponential Functions Calculator Demonstration Graph f (x) = 3 x. Then graph each of the following and describe the translations necessary to graph each from the parent function f (x) = 3 x. a g(x) = 3 x+1 b h(x) = 3 x 2 c j(x) = 3 x d k(x) = 3 x Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 10 / 17
Exponential Functions and Their Graphs (Part 1 of 2) Assignment pg. 226-227 #7-10, 11-15 odd, 16-20 Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 11 / 17
The Natural Base e The number e 2.718281828. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 12 / 17
The Natural Base e The number e 2.718281828. It is referred to as the natural base and is the most convenient choice for a base in many applications that incorporate exponential functions. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 12 / 17
The Natural Base e The number e 2.718281828. It is referred to as the natural base and is the most convenient choice for a base in many applications that incorporate exponential functions. f (x) = e x is called the natural exponential function. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 12 / 17
Graphing Natural Exponential Functions Practice Use a graphing calculator to construct a table of values for the function. Then sketch the graph of the function. a f (x) = 2e 0.24x b f (x) = 1 2 e 0.58x Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 13 / 17
Compound Interest Compound interest refers to interest being charged based on interest charged in the past. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 14 / 17
Compound Interest Compound interest refers to interest being charged based on interest charged in the past. Interest can be charged yearly, monthly, weekly, daily, hourly, or even continuously. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 14 / 17
Compound Interest Compound interest refers to interest being charged based on interest charged in the past. Interest can be charged yearly, monthly, weekly, daily, hourly, or even continuously. Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas. ( For n compoundings per year: A = P 1 + r ) nt n For continuous compounding: A = Pe rt Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 14 / 17
Compound Interest Practice A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a quarterly b monthly c continuously Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 15 / 17
Radioactive Decay Practice In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread highly toxic radioactive chemicals, such as plutonium, over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the model P = 10 ( ) t 1 24,100 2 which represents the amount of plutonium P that remains (from an initial amount of 10 pounds) after t years. Sketch the graph of this function over the interval from t = 0 to t = 10,000, where t = 0 represents 1986. How much of the 10 pounds will remain in the year 2010? How much of the 10 pounds will remain after 100,000 years? Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 16 / 17
Exponential Functions and Their Graphs (Part 2 of 2) Assignment pg. 226-227 #7-10, 11-15 odd, 16-20, 27-59 odd, 61-63 Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 17 / 17