exponential functions math secondary iv. topics calculation calculation growth & decay growth...
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Exponential FunctionsExponential Functions
Math Secondary IVMath Secondary IV
TopicsTopics
CalculationCalculationGrowth & DecayGrowth & DecayFactorFactorGraphGraphEquationEquationPoint of IntersectionPoint of IntersectionWord ProblemsWord Problems
CalculatorsCalculators
This topic will involve a lot of This topic will involve a lot of calculator use and specifically the calculator use and specifically the exponential keyexponential key
Usually it is a button yUsually it is a button yxx , x , xy y or ^or ^For example 25 = ?32177
410338673
Calculators (2Calculators (2ndnd Step) Step)
Add another step: 8 x 9Add another step: 8 x 944 5248852488 16 x 7216 x 7244
429981696429981696 42 x (2/3)42 x (2/3)88
Do 42 x open bracket 2/3 close bracket yDo 42 x open bracket 2/3 close bracket yxx 8 = ??8 = ??
1.64 (round off to two decimal places) 1.64 (round off to two decimal places)
Exercises Exercises
2.176 x 0.8152.176 x 0.81588
0.420.42 273 x 47273 x 471111
6.75 x 106.75 x 102020
0.82 x 0.00630.82 x 0.00634040
7.72 x 107.72 x 10-89-89
Work in class – Do Work in class – Do #1 a-j #1 a-j
Growth & DecayGrowth & DecayConsider the powers of 7 and 1/7Consider the powers of 7 and 1/77 = 77 = 700 = 1 = 17722 = 49 = 497733 = 343 = 343We note as the power increases We note as the power increases
value increases; we call this…value increases; we call this…growthgrowth
Growth & DecayGrowth & Decay
Consider 1/7Consider 1/700 = 1 = 1(1/7)(1/7)22 = 0.02 = 0.02(1/7)(1/7)33 = 0.0029 = 0.0029We note as the power increases the We note as the power increases the
value decreases; we call this… value decreases; we call this… decaydecay
Growth & DecayGrowth & Decay
In general, if the base or factor is In general, if the base or factor is greater than 1 we have growth.greater than 1 we have growth.
If the base or factor is between 0 and If the base or factor is between 0 and 1, we have decay.1, we have decay.
We note we do not use 0 and 1 or We note we do not use 0 and 1 or negative numbersnegative numbers
Work in class / homework – do #2 a-eWork in class / homework – do #2 a-e
FactorsFactors
We constantly need to see what We constantly need to see what factor or base we are using. factor or base we are using.
Some are easy Double = Some are easy Double = 22Half = Half = ½½By ten = By ten = 1010
FactorsFactors
Then there is per cent. Then there is per cent. If I haveIf I have
If I double it…If I double it… I would multiply by 2 I would multiply by 2
Other FactorsOther Factors
If I have a square and add 10% I If I have a square and add 10% I would multiply by? would multiply by?
1.1 (110%)1.1 (110%)We would get We would get
More ExamplesMore Examples
10% increase is a factor of 1.1 (1+.1)10% increase is a factor of 1.1 (1+.1)33% increase is a factor of 33% increase is a factor of 1.33 (1+.33)1.33 (1+.33)1% increase is a factor of 1% increase is a factor of 1.01 (1+.01)1.01 (1+.01)4.75 increase is a factor 4.75 increase is a factor 1.0475 (1+0.0475).1.0475 (1+0.0475).
Let’s Go the Other WayLet’s Go the Other Way
We can use the same logic for a We can use the same logic for a 10% decrease. 10% decrease is 10% decrease. 10% decrease is a factor of 0.9 (1-.1)a factor of 0.9 (1-.1)
39% decrease is a factor of 39% decrease is a factor of 0.61 (1-.39)0.61 (1-.39) Please note increase = up, Please note increase = up,
appreciation, interest and appreciation, interest and decrease = down, depreciationdecrease = down, depreciation
Work in class / homework do #3 Work in class / homework do #3 a-ja-j
Exponential FormulaExponential Formula
The exponential formula is y = S ● FThe exponential formula is y = S ● Fxx
Formula DefinedFormula DefinedS ParameterS Parameter: S is the : S is the startingstarting value value
Where does the function startWhere does the function start In other words what is the value of y In other words what is the value of y
when x=0when x=0
Exponential Formula DefinedExponential Formula Defined
F ParameterF Parameter – – F stands for factorF stands for factorwhat is the function increasing or what is the function increasing or
decreasing by?decreasing by? i.e., what is the ratio between the i.e., what is the ratio between the
value when x = 1 and x = 0value when x = 1 and x = 0 i.e. what is the value when x=1 i.e. what is the value when x=1
divideddivided by the value when x = 0. by the value when x = 0.
GraphsGraphs
Graph the following. State the S and F Graph the following. State the S and F parameters and whether the curve is a parameters and whether the curve is a growth or decay. growth or decay.
X 0 1 2 3 4 X 0 1 2 3 4 Y 2 6 18 54 162Y 2 6 18 54 162 S: x = 0 y = 2; S = ?S: x = 0 y = 2; S = ? S = 2 S = 2 F: x = 1 y = 6F: x = 1 y = 6 x = 0 y = 2; F = ?x = 0 y = 2; F = ? F = 6 / 2 = 3F = 6 / 2 = 3
Graph con’tGraph con’t
S = 2S = 2 F = 3 F = 3 Graph it!Graph it! GrowthGrowth Hence y = 2 ● 3Hence y = 2 ● 3xx
Work in class / Work in class / homework do #4 a-e; homework do #4 a-e; 2 per page!2 per page!
Homework SolutionsHomework Solutions
4a: S = 64a: S = 6F = 3F = 3GrowthGrowthGraph it on graph paper; two per Graph it on graph paper; two per
pagepage
Work in Class / HomeworkWork in Class / Homework
#5: y = 5 (2)#5: y = 5 (2)xx
Plot X and Y (use 1,2,3,&4)Quiz
ExercisesExercises
Give the formula to show the situation Give the formula to show the situation where you invest $1000 at 7% where you invest $1000 at 7% annually.annually.
After 20 years, how much do you After 20 years, how much do you have? have?
What is S and what is F?What is S and what is F?S = 1000 ; F = 1.07; Put in formula…S = 1000 ; F = 1.07; Put in formula…Where y is the money the investment Where y is the money the investment
is worth; x is the number of yearsis worth; x is the number of years
Work in Class / HomeworkWork in Class / Homework
#6 a – j #6 a – j State the Exponential Equation… 6aState the Exponential Equation… 6a6a) Y = 14 (6a) Y = 14 (½½))xx
6b) y = 1000 (1/100) 6b) y = 1000 (1/100) xx
SolutionSolution
y = 1000 ● (1.07)y = 1000 ● (1.07)xx
After 20 years your $1000 After 20 years your $1000 investment is worth… investment is worth…
Y = 1000 (1.07)Y = 1000 (1.07)2020
= $3869.68= $3869.68
More ExercisesMore Exercises
Radioactive elements decay (the atoms Radioactive elements decay (the atoms fall apart) in a set formula. The element fall apart) in a set formula. The element Pingdanga has a half life of a year. Give Pingdanga has a half life of a year. Give the formula if you start with 1000 kgthe formula if you start with 1000 kg
11stst step: Write down Exponential Formu. step: Write down Exponential Formu.Y = 1000 (Y = 1000 (½)½) xx
Y is the amount of PingdangaY is the amount of PingdangaX is the number of yearsX is the number of years
Solution Con’tSolution Con’t
In 20 years we would have… In 20 years we would have… Y = 1000 (Y = 1000 (½)½)2020
= 0.001 kg of Pingdanga= 0.001 kg of Pingdanga
Comparing InvestmentsComparing Investments
If you have two investments $1000 @ If you have two investments $1000 @ 20% and $2000 @ 5%, when will they 20% and $2000 @ 5%, when will they be worth the same? Plot points every be worth the same? Plot points every two years for eight years. What is the two years for eight years. What is the value of each investment after 50 value of each investment after 50 years?years?
11stst step… create two exponential step… create two exponential formulas (double the fun!) formulas (double the fun!)
Y = 1000 (1.2)Y = 1000 (1.2)XX y = 2000 (1.05) y = 2000 (1.05)XX
Create Table of Values for both…Create Table of Values for both…
Investment SolutionsInvestment Solutions
Investment AInvestment A
X 0 2 4 6 8 X 0 2 4 6 8
Y $Y $1000 $1440 $2073.60 $2985.98 1000 $1440 $2073.60 $2985.98 $4299.82$4299.82
Investment BInvestment B
X 0X 0 2 4 6 8 2 4 6 8
Y Y 2000 2205 2431.01 2680.19 2954.912000 2205 2431.01 2680.19 2954.91
Investment SolutionsInvestment Solutions
After 50 years?After 50 years? Investment AInvestment A1000 (1.2)1000 (1.2)5050
= $9100438.15= $9100438.15 Investment BInvestment B2000 (1.05)2000 (1.05)5050
= $22934.80= $22934.80
Car DepreciationCar Depreciation
Which car is worth more after five Which car is worth more after five years given the following values. You years given the following values. You are given the cost of the car and the are given the cost of the car and the depreciation (amount your car goes depreciation (amount your car goes down per year). down per year).
Car A: $5000 @ 7%Car A: $5000 @ 7%Car B: $6000 @ 8%Car B: $6000 @ 8%
Car SolutionCar Solution
Car A: 5000 (0.93)Car A: 5000 (0.93)55
= $3478.44= $3478.44Car B: $6000 (.92)Car B: $6000 (.92)55
= $3954.49= $3954.49
Work in Class / HomeworkWork in Class / Homework
#7-15…#7-15…Study GuideStudy GuideTest Test