Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By...
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Transcript of Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By...
![Page 1: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/1.jpg)
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Algebra 2Chapter 7
![Page 2: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/2.jpg)
This Slideshow was developed to accompany the textbookLarson Algebra 2By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. 2011 Holt McDougal
Some examples and diagrams are taken from the textbook.
Slides created by Richard Wright, Andrews Academy [email protected]
![Page 3: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/3.jpg)
7.1 Graph Exponential Growth Functions
![Page 4: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/4.jpg)
8-1 Exponential Growth
![Page 5: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/5.jpg)
WRIGHT
![Page 6: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/6.jpg)
7.1 Graph Exponential Growth Functions Exponential Function
y = bx
Base (b) is a positive number other than 1
y = 2x
![Page 7: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/7.jpg)
![Page 8: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/8.jpg)
7.1 Graph Exponential Growth Functions y = a · 2x
y-intercept = ax-axis is the asymptote of graph
![Page 9: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/9.jpg)
7.1 Graph Exponential Growth Functions Exponential Growth Function
y = a · bx – h + k
To graphStart with y = bx
Multiply y-coordinates by aMove up k and right h(or make table of values)
Properties of the graph
y-intercept = a (if h and k=0) y = k is asymptote Domain is all real numbers Range
y > k if a > 0y < k if a < 0
![Page 10: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/10.jpg)
7.1 Graph Exponential Growth Functions Graph y = 3 · 2x – 3 – 2
![Page 11: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/11.jpg)
7.1 Graph Exponential Growth Functions Exponential Growth Model (word problems)
y = a(1 + r)t
○ y = current amount○ a = initial amount○ r = growth percent○ t = time
![Page 12: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/12.jpg)
7.1 Graph Exponential Growth Functions Compound Interest
○ A = current amount○ P = principle (initial amount)○ r = percentage rate○ n = number of times compounded per year○ t = time in years
![Page 13: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/13.jpg)
7.1 Graph Exponential Growth Functions If you put $200 into a CD (Certificate of Deposit) that earns 4%
interest, how much money will you have after 2 years if you compound the interest monthly? daily?
482 #1, 5, 7, 9, 13, 17, 19, 21, 27, 29, 35, 37 + 3 = 15 total
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Quiz 7.1 Homework Quiz
![Page 15: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/15.jpg)
7.2 Graph Exponential Decay Functions Exponential Decay
y = a·bx
a > 00 < b < 1
Follows same rules as growthy-intercept = ay = k is asymptotey = a · bx – h + k
y = (½)x
![Page 16: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/16.jpg)
7.2 Graph Exponential Decay Functions Graph y = 2 · (½)x + 3 – 2
![Page 17: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/17.jpg)
7.2 Graph Exponential Decay Functions Exponential Decay Model (word problems)
y = a(1 - r)t
○ y = current amount○ a = initial amount○ r = decay percent○ t = time
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7.2 Graph Exponential Decay Functions A new car cost $23000. The value decreases by 15%
each year. Write a model of this decay. How much will the car be worth in 5 years? 10 years?
489 #1, 3, 5, 7, 11, 15, 17, 19, 27, 31, 33 + 4 = 15 total
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Quiz 7.2 Homework Quiz
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7.3 Use Functions Involving e In math, there are some special numbers like π or i
Today we will learn about e
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7.3 Use Functions Involving e e
Called the natural baseNamed after Leonard Euler who discovered it○ (Pronounced “oil-er”)
Found by putting really big numbers into = 2.718281828459…Irrational number like π
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7.3 Use Functions Involving e Simplifying natural base
expressionsJust treat e like a regular
variable (2𝑒−5 𝑥 )−2
![Page 23: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/23.jpg)
7.3 Use Functions Involving e Evaluate the natural base expressions using your calculator
e3
e-0.12
![Page 24: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/24.jpg)
7.3 Use Functions Involving e To graph make a table of
values f(x) = a·erx
a > 0If r > 0 growthIf r < 0 decay
Graph y = 2e0.5x
![Page 25: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/25.jpg)
7.3 Use Functions Involving e Compound Interest
○ A = current amount○ P = principle (initial amount)○ r = percentage rate○ n = number of times compounded per year○ t = time in years
Compounded continuouslyA = Pert
![Page 26: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/26.jpg)
7.3 Use Functions Involving e 495 #1-49 every other odd, 55, 57, 61 + 4 = 20 total
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Quiz 7.3 Homework Quiz
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7.4 Evaluate Logarithms and Graph Logarithmic Functions
Definition of Logarithm with Base b
Read as “log base b of y equals x”
Rewriting logarithmic equations log3 9 = 2 log8 1 = 0 log5(1 / 25) = -2
![Page 29: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/29.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
Special Logslogb 1 = 0
logb b = 1 Evaluate
log4 64
log2 0.125
log1/4 256
![Page 30: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/30.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
Using a calculator Common Log (base 10)
log10 x = log xFind log 12
Natural Log (base e)loge x = ln xFind ln 2
![Page 31: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/31.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
When the bases are the same, the base and the log cancel
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7.4 Evaluate Logarithms and Graph Logarithmic Functions
Finding Inverses of Logs y = log8 x x = log8 y Switch x and y y = 8x Rewrite to solve for y
To graph logsFind the inverseMake a table of values for the inverseGraph the log by switching the x and y coordinates of the inverse.
![Page 33: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/33.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
Properties of graphs of logs
y = logb (x – h) + k
x = h is vert. asymptoteDomain is x > hRange is all real numbersIf b > 1, graph risesIf 0 < b < 1, graph falls
![Page 34: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/34.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
Graph y = log2 x
Inversex = log2 yy = 2x
x y
-3 1/8
-2 ¼
-1 ½
0 1
1 2
2 4
3 8
![Page 35: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/35.jpg)
7.4 Evaluate Logarithms and Graph Logarithmic Functions
503 #3, 5-49 every other odd, 59, 61 + 5 = 20 total
![Page 36: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/36.jpg)
Quiz 7.4 Homework Quiz
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7.5 Apply Properties of Logarithms Product Property
Quotient Property
Power Property
![Page 38: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/38.jpg)
7.5 Apply Properties of Logarithms Use log9 5 = 0.732 and log9 11 = 1.091 to find
log9 55
log9 25
![Page 39: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/39.jpg)
7.5 Apply Properties of Logarithms Expand: log5 2x6 Condense: 2 log3 7 – 5 log3 x
![Page 40: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/40.jpg)
7.5 Apply Properties of Logarithms Change-of-Base Formula
Evaluate log4 8
510 #3-31 every other odd, 33-43 odd, 47, 51, 55, 59, 63, 71, 73 + 4 = 25 total
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Quiz 7.5 Homework Quiz
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7.6 Solve Exponential and Logarithmic Equations
Solving Exponential EquationsMethod 1) if the bases are equal, then exponents are equal
24x = 32x-1
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7.6 Solve Exponential and Logarithmic Equations Solving Exponential
EquationsMethod 2) take log of both
sides4x = 15
5x+2 + 3 = 25
![Page 44: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/44.jpg)
7.6 Solve Exponential and Logarithmic Equations
Solving Logarithmic EquationsMethod 1) if the bases are equal, then logs are equal
log3 (5x – 1) = log3 (x + 7)
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7.6 Solve Exponential and Logarithmic Equations
Solving Logarithmic EquationsMethod 2) exponentiating both sides○ Make both sides exponents with the base of the log
log4 (x + 3) = 2
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7.6 Solve Exponential and Logarithmic Equations
519 #3-43 every other odd, 49, 53, 55, 57 + 5 = 20 total
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Quiz 7.6 Homework Quiz
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7.7 Write and Apply Exponential and Power Functions
Just as 2 points determine a line, so 2 points will determine an exponential equation.
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7.7 Write and Apply Exponential and Power Functions
Exponential Functiony = a bx
If given 2 pointsFill in both points to get two equationsSolve for a and b by substitution
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7.7 Write and Apply Exponential and Power Functions
Find the exponential function that goes through (-1, 0.0625) and (2, 32)
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7.7 Write and Apply Exponential and Power Functions
Steps if given a table of valuesFind ln y of all pointsGraph ln y vs xDraw the best fit straight linePick two points on the line and find equation of line (remember to use ln y
instead of just y)Solve for y
OR use the ExpReg feature on a graphing calculatorEnter points in STAT EDITGo to STAT CALC ExpReg Enter Enter
![Page 52: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/52.jpg)
7.7 Write and Apply Exponential and Power Functions
Writing a Power Functiony = a xb
Steps are the same as for exponential functionFill in both points to get two equationsSolve for a and b by substitution
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7.7 Write and Apply Exponential and Power Functions
Write power function through (3, 8) and (9, 12)
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7.7 Write and Apply Exponential and Power Functions
Steps if given a table of valuesFind ln y and ln x of all pointsGraph ln y vs ln xDraw the best fit straight linePick two points on the line and find equation of line (remember to use ln y
and ln x instead of just y)Solve for y
OR use the PwrReg feature on a graphing calculatorEnter points in STAT EDITGo to STAT CALC PwrReg Enter Enter
![Page 55: Algebra 2 Chapter 7. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649d6f5503460f94a51b79/html5/thumbnails/55.jpg)
7.7 Write and Apply Exponential and Power Functions
533 #3, 7, 11, 13, 15, 19, 23, 27, 33, 35 + 5 = 15 total
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Quiz 7.7 Homework Quiz
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7.Review 543 choose 20