1. 1/20/2016 Integrated Math 10 - Santowski 1 Lesson 13 Introducing Exponential Equations Integrated Math 10 - Santowski

2. Comparing Linear & Exponential Models Data set #1 – Can this data be modeled with a linear relation? Why/why not? How do you know? Data set #2 - Can this data be modeled with a linear relation? Why/why not? How do you know? 1/20/2016 Integrated Math 10 - Santowski 2 x0123457 y1624365481121.5273.375 x0123457 y5101520253040

3. HINT Any obvious number patterns in the data?? 1/20/2016 Integrated Math 10 - Santowski 3

4. Comparing Linear & Exponential Models Data set #3 – Can this data be modeled with a linear relation? Why/why not? How do you know? Data set #4 - Can this data be modeled with a linear relation? Why/why not? How do you know? 1/20/2016 Integrated Math 10 - Santowski 4 x0123456 y5075112.5168.75253.13379.69569.53 x0123456 y57.7510.513.251618.7521.5

5. Comparing Linear & Exponential Models Data set #5 – Can this financial data be modeled with a linear relation? Why/why not? How do you know? What equation summarizes the data relationship and what do the #’s in the eqn represent? Data set #6 - Can this financial data be modeled with a linear relation? Why/why not? How do you know? What equation summarizes the data relationship and what do the #’s in the eqn represent? 1/20/2016 Integrated Math 10 - Santowski 5 x0123457 y50006000720086401036812441.617915.90 x0123457 y500060007000800090001000012000

6. Summary What NEW pattern/relationships have you seen in the data sets? How can we write equations for our data that reflect our new patterns/relationships? 1/20/2016 Integrated Math 10 - Santowski 6

7. Summary How can we write equations for our data that reflect our new patterns/relationships? If you have correctly answered this question, go to slides #12 - 16 How can we write equations for our data that reflect our new patterns/relationships? If you have NOT correctly answered this question, go to slides #9 - #11 1/20/2016 Integrated Math 10 - Santowski 7

8. 1/20/2016 Integrated Math 10 - Santowski 8 Additional data sets (if needed)

9. Exploring Exponential Equations Data set #A – This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? Data set #B - This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? 1/20/2016 Integrated Math 10 - Santowski 9 x0123456 y7292438127931 x0123456 y1248163264

10. Exploring Exponential Equations Data set #C – This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? Data set #D - This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? 1/20/2016 Integrated Math 10 - Santowski 10 x0123456 y100125156.25195.31244.14305.18381.47 x0123456 y0.30.61.22.44.89.619.2

11. Exploring Exponential Equations Data set #E – This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? Data set #F - This data can be modeled with a exponential relation. How do you know? What “equation” summarizes the data relationship? 1/20/2016 Integrated Math 10 - Santowski 11 x0123456 y3601204013.334.441.480.494 x0123456 y50301810.86.483.892.33

12. 1/20/2016 Integrated Math 10 - Santowski 12 Application of Exponential Models

13. Modeling Example #1 Investment data – Mr. S has invested some money for Andrew’s post- secondary education (not too hopeful for an athletic scholarship for my son!!!!) (a) How can you analyze the numeric data (no graphs) to conclude that the data is exponential  i.e how do you know the data is exponential rather than linear? (b) Graph the data on a scatter plot (c) Write an equation to model the data. Define your variables carefully. 1/20/2016 Integrated Math 10 - Santowski 13 Time (years)012345678 Value of investment (000’s $) 88.4808.9899.52810.00010.70611.34812.02912.751

14. Modeling Example #2 The following data table shows the relationship between the time (in hours after a rain storm in Manila) and the number of bacteria (#/mL of water) in water samples from the Pasig River: (a) How can you analyze the numeric data (no graphs) to conclude that the data is exponential  i.e how do you know the data is exponential rather than linear? (b) Graph the data on a scatter plot (c) Write an equation to model the data. Define your variables carefully. 1/20/2016 Integrated Math 10 - Santowski 14 Time (hrs) 012345678 # of Bacteria 1001963958061570315462151260025300

15. Modeling Example #3 The value of Mr. S’s car is depreciating over time. I bought the car new in 2002 and the value of my car (in thousands) over the years has been tabulated below: (a) How can you analyze the numeric data (no graphs) to conclude that the data is exponential  i.e how do you know the data is exponential rather than linear? (b) Graph the data on a scatter plot (c) Write an equation to model the data. Define your variables carefully. 1/20/2016 Integrated Math 10 - Santowski 15 Year200220032004200520062007200820092010 Value403632.429.226.223.621.319.117.2

16. Modeling Example #4 The following data table shows the historic world population since 1950: (a) How can you analyze the numeric data (no graphs) to conclude that the data is exponential  i.e how do you know the data is exponential rather than linear? (b) Graph the data on a scatter plot (c) Write an equation to model the data. Define your variables carefully. 1/20/2016 Integrated Math 10 - Santowski 16 Year195019601970198019901995200020052010 Pop (in millions) 2.563.043.714.455.295.7806.096.476.85

17. SUMMARY Write a SINGLE equation that summarizes the data relationships you have investigated this lesson. 1/20/2016 Integrated Math 10 - Santowski 17

18. 1/20/2016 Integrated Math 10 - Santowski 18 SUMMARY  Exponential Growth Equations In general, the algebraic model for exponential growth is y = c(a) x where a is referred to as the growth rate (provided that a > 1) and c is the initial amount present and x is the number of increases given the growth rate conditions. All equations in this section are also written in the form y = c(1 + r) x where c is a constant, r is a positive rate of change and 1 + r > 1, and x is the number of increases given the growth rate conditions.

19. 1/20/2016 Integrated Math 10 - Santowski 19 SUMMARY  Exponential Decay Equations In general, the algebraic model for exponential decay is y = c(a) x where a is referred to as the decay rate (and a is < 1) and c is the initial amount present. All equations in this section are in the form y = c(1 + r) x or y = ca x, where c is a constant, r is a rate of change (this time negative as we have a decrease so 1 + r < 1), and x is the number of increases given the rate conditions