L1.4 A2 Exponential Functions - Part 5
Completion requirements
Unit 1
Functions
Exponential Growth
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»c«/mi»«mo»§#62;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«mo»=«/mo»«msup»«mi»c«/mi»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» is an increasing function. As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» increases, so does «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
Examples of exponential growth are population growth, doubling time for an investment, or the spread of a virus through a population.
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»c«/mi»«mo»§#62;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«mo»=«/mo»«msup»«mi»c«/mi»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» is an increasing function. As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» increases, so does «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
Examples of exponential growth are population growth, doubling time for an investment, or the spread of a virus through a population.
Exponential Decay
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#60;«/mo»«mi»c«/mi»«mo»§#60;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«mo»=«/mo»«msup»«mi»c«/mi»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» is a decreasing function. As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» increases, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» decreases.
Examples of exponential decay are the depreciation of the value of a car, the half-life of a radioactive substance, or the effects of an antibiotic on a population of bacteria cells.
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#60;«/mo»«mi»c«/mi»«mo»§#60;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«mo»=«/mo»«msup»«mi»c«/mi»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» is a decreasing function. As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» increases, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» decreases.
Examples of exponential decay are the depreciation of the value of a car, the half-life of a radioactive substance, or the effects of an antibiotic on a population of bacteria cells.
Under the right conditions, a certain type of bacteria’s cell count will increase exponentially every day («math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»24«/mn»«/mstyle»«/math» hour period).
Use the table of values to answer the following questions. The variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» represents the number of days, and the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mstyle»«/math» is the number of cells, in hundreds, after «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» days.
It will take approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo».«/mo»«mn»26«/mn»«/mrow»«/mstyle»«/math» days for the bacteria count to reach «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo»§#160;«/mo»«mn»200«/mn»«/mstyle»«/math».
Use the table of values to answer the following questions. The variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» represents the number of days, and the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mstyle»«/math» is the number of cells, in hundreds, after «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» days.
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathcolor=¨#FFFFFF¨»t«/mi»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathcolor=¨#FFFFFF¨»n«/mi»«mfenced mathcolor=¨#FFFFFF¨»«mi»t«/mi»«/mfenced»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»9«/mn»«/mstyle»«/math» |
a.
What are the domain and range of the function?
b.
Write an exponential growth function that relates the number of cells to the time in days.
c.
Determine approximately how many days it would take for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo»§#160;«/mo»«mn»200«/mn»«/mstyle»«/math» cells to be present.
a.
The domain is the time passed, in days.
The range is the number of cells, in hundreds.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»t«/mi»«mo»§#8739;«/mo»«mi»t«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»t«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
The range is the number of cells, in hundreds.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»§#8739;«/mo»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»§#8805;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»W«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
b.
For each increase of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mstyle»«/math» increases by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math». This means the base will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math». The function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«msup»«mn»3«/mn»«mi»t«/mi»«/msup»«/mrow»«/mstyle»«/math».
c.
The number of cells, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mstyle»«/math», is in hundreds, so substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mstyle»«/math».
This equation can be solved graphically. Graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»12«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«msup»«mn»3«/mn»«mi»t«/mi»«/msup»«/mstyle»«/math», and determine any points of intersection.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»12«/mn»«mo»=«/mo»«msup»«mn»3«/mn»«mi»t«/mi»«/msup»«/mrow»«/mstyle»«/math»
This equation can be solved graphically. Graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»12«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«msup»«mn»3«/mn»«mi»t«/mi»«/msup»«/mstyle»«/math», and determine any points of intersection.
It will take approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo».«/mo»«mn»26«/mn»«/mrow»«/mstyle»«/math» days for the bacteria count to reach «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo»§#160;«/mo»«mn»200«/mn»«/mstyle»«/math».