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Exponential Functions Math Secondary IV

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Topics Calculation Calculation Growth & Decay Growth & Decay Factor Factor Graph Graph Equation Equation Point of Intersection Point of Intersection Word Problems Word Problems

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Calculators This topic will involve a lot of calculator use and specifically the exponential key This topic will involve a lot of calculator use and specifically the exponential key Usually it is a button y x, x y or ^ Usually it is a button y x, x y or ^ For example 2 5 = ? 32 17 7 410338673

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Calculators (2 nd Step) Add another step: 8 x 9 4 Add another step: 8 x 9 4 52488 52488 16 x 72 4 16 x 72 4 429981696 429981696 42 x (2/3) 8 42 x (2/3) 8 Do 42 x open bracket 2/3 close bracket y x 8 = ?? Do 42 x open bracket 2/3 close bracket y x 8 = ?? 1.64 (round off to two decimal places) 1.64 (round off to two decimal places)

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Exercises 2.176 x 0.815 8 2.176 x 0.815 8 0.42 0.42 273 x 47 11 273 x 47 11 6.75 x 10 20 6.75 x 10 20 0.82 x 0.0063 40 0.82 x 0.0063 40 7.72 x 10 -89 7.72 x 10 -89 Work in class – Do #1 a-j Work in class – Do #1 a-j

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Growth & Decay Consider the powers of 7 and 1/7 Consider the powers of 7 and 1/7 7 = 7 0 = 1 7 = 7 0 = 1 7 2 = 49 7 2 = 49 7 3 = 343 7 3 = 343 We note as the power increases value increases; we call this… We note as the power increases value increases; we call this… growth growth

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Growth & Decay Consider 1/7 0 = 1 Consider 1/7 0 = 1 (1/7) 2 = 0.02 (1/7) 2 = 0.02 (1/7) 3 = 0.0029 (1/7) 3 = 0.0029 We note as the power increases the value decreases; we call this… We note as the power increases the value decreases; we call this… decay decay

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Growth & Decay In general, if the base or factor is greater than 1 we have growth. In general, if the base or factor is greater than 1 we have growth. If the base or factor is between 0 and 1, we have decay. If the base or factor is between 0 and 1, we have decay. We note we do not use 0 and 1 or negative numbers We note we do not use 0 and 1 or negative numbers Work in class / homework – do #2 a-e Work in class / homework – do #2 a-e

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Factors We constantly need to see what factor or base we are using. We constantly need to see what factor or base we are using. Some are easy Double = Some are easy Double = 2 Half = Half = ½ By ten = By ten = 10 10

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Factors Then there is per cent. Then there is per cent. If I have If I have If I double it… If I double it… I would multiply by 2 I would multiply by 2

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Other Factors If I have a square and add 10% I would multiply by? If I have a square and add 10% I would multiply by? 1.1 (110%) 1.1 (110%) We would get We would get

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More 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).

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Lets Go the Other Way We can use the same logic for a 10% decrease. 10% decrease is a factor of 0.9 (1-.1) We can use the same logic for a 10% decrease. 10% decrease is 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, appreciation, interest and decrease = down, depreciation Please note increase = up, appreciation, interest and decrease = down, depreciation Work in class / homework do #3 a-j Work in class / homework do #3 a-j

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Exponential Formula The exponential formula is y = S F x The exponential formula is y = S F x Formula Defined Formula Defined S Parameter: S is the starting value S Parameter: S is the starting value Where does the function start In other words what is the value of y when x=0 In other words what is the value of y when x=0

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Exponential Formula Defined F Parameter – F Parameter – F stands for factor F stands for factor what is the function increasing or decreasing by? what is the function increasing or decreasing by? i.e., what is the ratio between the value when x = 1 and x = 0 i.e., what is the ratio between the value when x = 1 and x = 0 i.e. what is the value when x=1 divided by the value when x = 0. i.e. what is the value when x=1 divided by the value when x = 0.

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Graphs Graph the following. State the S and F parameters and whether the curve is a growth or decay. Graph the following. State the S and F parameters and whether the curve is a growth or decay. X 0 1 2 3 4 X 0 1 2 3 4 Y 2 6 18 54 162 Y 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 = 6 F: x = 1 y = 6 x = 0 y = 2; F = ? x = 0 y = 2; F = ? F = 6 / 2 = 3 F = 6 / 2 = 3

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Graph cont S = 2 S = 2 F = 3 F = 3 Graph it! Graph it! Growth Growth Hence y = 2 3 x Hence y = 2 3 x Work in class / homework do #4 a-e; 2 per page! Work in class / homework do #4 a-e; 2 per page!

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Homework Solutions 4a: S = 6 4a: S = 6 F = 3 F = 3 Growth Growth Graph it on graph paper; two per page Graph it on graph paper; two per page

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Work in Class / Homework #5: y = 5 (2) x #5: y = 5 (2) x Plot X and Y (use 1,2,3,&4) Quiz

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Exercises Give the formula to show the situation where you invest $1000 at 7% annually. Give the formula to show the situation where you invest $1000 at 7% annually. After 20 years, how much do you have? After 20 years, how much do you 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 is worth; x is the number of years Where y is the money the investment is worth; x is the number of years

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Work in Class / Homework #6 a – j #6 a – j State the Exponential Equation… 6a State the Exponential Equation… 6a 6a) Y = 14 (½) x 6a) Y = 14 (½) x 6b) y = 1000 (1/100) x 6b) y = 1000 (1/100) x

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Solution y = 1000 (1.07) x y = 1000 (1.07) x After 20 years your $1000 investment is worth… After 20 years your $1000 investment is worth… Y = 1000 (1.07) 20 Y = 1000 (1.07) 20 = $3869.68 = $3869.68

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More Exercises Radioactive elements decay (the atoms fall apart) in a set formula. The element Pingdanga has a half life of a year. Give the formula if you start with 1000 kg Radioactive elements decay (the atoms fall apart) in a set formula. The element Pingdanga has a half life of a year. Give the formula if you start with 1000 kg 1 st step: Write down Exponential Formu. 1 st step: Write down Exponential Formu. Y = 1000 (½) x Y = 1000 (½) x Y is the amount of Pingdanga Y is the amount of Pingdanga X is the number of years X is the number of years

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Solution Cont In 20 years we would have… In 20 years we would have… Y = 1000 (½) 20 Y = 1000 (½) 20 = 0.001 kg of Pingdanga = 0.001 kg of Pingdanga

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Comparing Investments If you have two investments $1000 @ 20% and $2000 @ 5%, when will they be worth the same? Plot points every two years for eight years. What is the value of each investment after 50 years? If you have two investments $1000 @ 20% and $2000 @ 5%, when will they be worth the same? Plot points every two years for eight years. What is the value of each investment after 50 years? 1 st step… create two exponential formulas (double the fun!) 1 st step… create two exponential formulas (double the fun!) Y = 1000 (1.2) X y = 2000 (1.05) X Y = 1000 (1.2) X y = 2000 (1.05) X Create Table of Values for both… Create Table of Values for both…

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Investment Solutions Investment A Investment A X 0 2 4 6 8 X 0 2 4 6 8 Y $ 1000 $1440 $2073.60 $2985.98 $4299.82 Y $ 1000 $1440 $2073.60 $2985.98 $4299.82 Investment B Investment B X 0 2 4 6 8 X 0 2 4 6 8 Y 2000 2205 2431.01 2680.19 2954.91 Y 2000 2205 2431.01 2680.19 2954.91

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Investment Solutions After 50 years? After 50 years? Investment A Investment A 1000 (1.2) 50 1000 (1.2) 50 = $9100438.15 = $9100438.15 Investment B Investment B 2000 (1.05) 50 2000 (1.05) 50 = $22934.80 = $22934.80

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Car Depreciation Which car is worth more after five years given the following values. You are given the cost of the car and the depreciation (amount your car goes down per year). Which car is worth more after five years given the following values. You are given the cost of the car and the depreciation (amount your car goes down per year). Car A: $5000 @ 7% Car A: $5000 @ 7% Car B: $6000 @ 8% Car B: $6000 @ 8%

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Car Solution Car A: 5000 (0.93) 5 Car A: 5000 (0.93) 5 = $3478.44 = $3478.44 Car B: $6000 (.92) 5 Car B: $6000 (.92) 5 = $3954.49 = $3954.49

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Work in Class / Homework #7-15… #7-15… Study Guide Study Guide Test Test

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