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