7.1 Characteristics of Exponential Functions.notebook. Chapter 7: Exponential Functions

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1 Chapter 7: Exponential Functions 1

2 Chapter Characteristics of Exponential Functions Pages Investigating Exponential Functions: 1. Complete the following table using and sketch on the axis provided. x y 2

3 2. Complete the following table using your results. Domain Range x intercept y intercept asymptote exponential function a function of the form is a variable. Exponential growth if c > 1 Exponential decay if 0 < c < 1 Standard table for any exponential function: where c is a constant (c > 0) and x x y c 3

Then identify the following: the domain and the range the x intercept and the

4 Example 1 Page 336 Analyse the Graph of an Exponential Function Graph each exponential function. Then identify the following: the domain and the range the x intercept and the y intercept if they exist whether the graph represents an increasing or a decreasing function. the equation of the horizontal asymptote. 4

5 Function Domain Range x intercept y intercept a) b) growth or decay asymptote 5

6 Example 1: Your Turn Page 338 Graph the exponential function y = 3 x without technology. Identify the following: the domain and range the x intercept and the y intercept, if they exist whether the graph represents an increasing or a decreasing function the equation of the horizontal asymptote Verify your results using graphing technology. Answer 6

Applications of Exponential Equations: General Formula: (variables change depending upon context) A(t) > final amount > starting or initial amount c > base Growth: Decay: doubling, c =

7 Applications of Exponential Equations: General Formula: (variables change depending upon context) A(t) > final amount > starting or initial amount c > base Growth: Decay: doubling, c = 2, tripling, c = 3, etc. half life, c = 1/2 per cent growth, c = 1 + % as decimal per cent decay, c = 1 % as decimal t > elapsed time d > time it takes to double, triple, half life, etc 7

8 Compound interest formula: > annual interest rate > number of compounding periods in a year. Examples: semi annually 2 quarterly 4 monthly 12 represent the same values as previous formula. 8

9 Some examples: 1. An initial count of bacteria shows It triples every 25 hours. a) Write the equation to model this situation. b) How many bacteria will there be in 4 days? 9

10 2. The intensity of the light below the surface of a lake is reduced by 4 % for every meter below the surface. a) Write the equation to model this situation. b) What percent of the original intensity remains 10 m below the surface? 3. John buys a new vehicle for $ It depreciates by 15 % each year. a) Write an equation that models this situation. b) What is the vehicle worth in 7 years? 10

11 4. The world's population is growing exponentially. In 1970 it was about 3.6 billion. If the populat increased at 2 % per year since then: a) What will be the population in 2020? b) When was the population 5 billion? 5. The half life of a certain isotope is 2 days. How much will be left from a mass of 500 g after: a) 6 days? b) 2 weeks? 11

12 6. Each of the following situations can be modeled using an exponential function. Indicate which situations model exponential growth and which situations model exponential decay. Also, give the value of c. a) The summertime population of gophers in a field increases by 10 % every year. b) The value of a new car decreases at a rate of rate of 15 % per year. c) The half life of Uranium 232 is 68.9 years. d) A type of bacterium doubles every 45 minutes. e) Money invested in a bank earns 2.5 % compounded annually. 12

13 #15 Page 345 modified Investing $1, compounded for the different times given at 100% interest annually. Complete the following chart: Compounding Periods Amount($) 13

,as n gets increasingly large, approaches 2.718 281 828 459 045 235.

14 ,as n gets increasingly large, approaches This is an irrational number just like Any situations involving continuous growth or decay in physics, chemistry, biology uses this number. It has so many applications it is given the symbol which is found on your calculator. 14

Exponential Functions with Base e

Exponential Functions with Base e Any positive number can be used as the base for an exponential function, but some bases are more useful than others. For instance, in computer science applications, the