1. Topic 6: Exponential Functions

2. Exploring Exponential Models

3. Exponential Functions An exponential function is a function with the general form: In an exponential function, the base b is a constant. The exponent x is the independent variable with domain the set of real numbers.

4. x f ( x ) -3 -2 0 1 2 3 What is the graph of ?

5. xf (x) -3 -2 0 1 2 3 What is the graph of f(x) = 3 x ? f(x) = 3 x

6. xf (x) -3 -2 0 1 2 3

7. xf (x) -3 -2 0 1 2 3 What is the graph of f(x) = -2 x ? f(x) = -2 x

8. Transformation of Exponentials Transformation Positive Leading Coefficient Negative Leading Coefficient

9. Vertical Shifts Shift Positive Negative

10. Horizontal Shifts Shift Positive Negative

11. Growth vs. Decay Exponential growth: as the value of x increases, the value of y increases Exponential decay: as the value of x increase, the value of y decreases, approaching zero.

12. Exponential Functions Coefficient Growth Decay

13. Identify each function or situation as an example of exponential growth or decay. Explain how you know.

14. Exponential Growth and Decay For exponential functions,the value of b is the growth or decay factor. You can model exponential growth or decay with this function: Amount after t time periods. Initial amount Rate of growth (r > 0) or decay (r< 0) # of time periods

15. You invest $1000 in a savings account at the end of 9 th grade. The account pays 5% annual interest. How much money will be in the account after six years?

16. Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no additional deposits or withdrawals, how many years will it take for the account to grow to at least $1500?

17. Rate of Change To model a situation using an exponential function of the form, you need to find the growth or decay factor b. If you know y-values for two consecutive x-values, you can find the rate of change r, and then find b using and

18. The table shows the world population of the Iberian lynx in 2013 and 2014. If this trend continues and the population is decreasing exponentially, how many Iberian lynx will there be in 2016? World Population of Iberian Lynx Year20132014 Population150120

19. Geometric & Arithmetic Sequences

20. Geometric vs. Arithmetic

22. Geometric Sequences What is the next term? What is the 7 th term? 1. 7, 49, 343…. 2. 0.5, -1, 2, -4…. 3. 1000, 250, 62.5, 15.625….

23. Arithmetic Sequences What is the next term? What is the 7 th term? 1. 1, 4, 7, 10…. 2. -6, -11, -16, -21 …. 3. 2.5, 5, 7.5, 9…

24. Explicit vs. Recursive Formula Recursive: formula used to find the next term of the sequence using one or more preceding terms of the sequence. Explicit: formula used to find the nth term of the sequence using one or more preceding terms of the sequence.

25. a n = current terma n -1 = pervious term a = 1 st termn = term number r = common ratio (amount multiplied each time)

26. a n = current terma n -1 = pervious term a = 1 st termn = term number d = common difference (amount added/subtracted each time)

27. Geometric Sequences What is the explicit formula for the nth term? What is recursive formula for the 7 th term? 1. 7, 49, 343…. 2. 0.5, -1, 2, -4…. 3. 1000, 250, 62.5, 15.625….

28. Arithmetic Sequences What is the explicit formula for the nth term? What is recursive formula for the 8 th term? 1. 1, 4, 7, 10…. 2. -6, -11, -16, -21 …. 3. 2.5, 5, 7.5, 9…

29. Scatterplots, Correlation & Regression

30. Explanatory & Response Variables Response Variables (Dependent Variables) Accident death rate Life expectancy Algebra 1 class grades Explanatory Variables (Independent Variables ) Car weight Number of cigarettes smoked Number of hours studied

31. Scatterplots A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Each individual in the data appears as a point on the graph.

32. How to Make a Scatterplot 1.Decide which variable should go on each axis. Remember, the eXplanatory variable goes on the X-axis! 2.Label and scale your axes. 3.Plot individual data values.

33. Displaying Relationships: Scatterplots Make a scatterplot of the relationship between body weight andpack weight. Body weight is our eXplanatory variable. Body weight (lb) 120187109103131165158116 Backpack weight (lb) 2630262429353128

34. Words That Describe… Direction (slope) – Positive or Negative Form – Linear, quadratic, cubic, exponential, curved, non-linear, etc. Strength – Strong, weak, somewhat strong, very weak,moderately strong, etc.

35. Describe this Scatterplot

37. What is Correlation? Correlation describes what percent ofvariation in y is ‘explained’ by x. Correlation values are between -1 and 1. Correlation is abbreviated: r The strength of the linear relationshipincreases as r moves away from 0 towards -1or 1.

38. Scatterplots and Correlation

39. What does “r” mean? R ValueStrength Perfectly linear; negative -0.75Strong negative relationship -0.50Moderately strong negative relationship -0.25Weak negative relationship 0nonexistent 0.25Weak positive relationship 0.50Moderately strong positive relationship 0.75Strong positive relationship 1Perfectly linear; positive

40. How strong is the correlation? Is it positive or negative? 0.235 -0.456 0.975 -0.784

41. Describe the scatterplot below. Be sure to estimate the correlation.

42. As the number of boats registered in Floridaincreases so does the number of manateeskilled by boats. This relationship is a strong, positive linearrelationship. The estimated correlation is approximately r=0.85. **Answers between 0.75-0.95 would beacceptable.

43. Correlation: Highly Effected By Outliers

44. Regression Lines A regression line summarizes the relationship between two variables, but only in settings where oneof the variables helps explain or predict the other. A regression line is a line that describes how a response variable y changes as an explanatory variable x changes.

45. Regression Lines Regression lines are used to conductanalysis: Professional sports teams use player’svital stats (40 yard dash, height, weight)to predict success Macy’s uses shipping, sales and inventorydata predict future sales. MDCPS uses regression equations toevaluate teachers based on student testdata.

46. Regression Line Equation

48. Interpreting Linear Regression Y-intercept: Slope:

49. Interpreting Regression Lines & Predicted Value Data on the IQ test scores and reading test scores for agroup of ninth-grade students resulted in the followingregression line:predicted reading score = −33.4 + 0.882(IQ score)(a) What’s the slope of this line? Interpret this value in context.

50. Interpreting Regression Lines & Predicted Value Data on the IQ test scores and reading test scores for agroup of ninth-grade students resulted in the followingregression line:predicted reading score = −33.4 + 0.882(IQ score)(b) What’s the y-intercept? Explain why the value of the intercept is not statistically meaningful.

51. Interpreting Regression Lines & Predicted Value Data on the IQ test scores and reading test scores for agroup of ninth-grade students resulted in the followingregression line:predicted reading score = −33.4 + 0.882(IQ score)(c) Find the predicted reading scores for two children with IQ scores of 90 and 130, respectively.

54. Linear or Exponential? Word of mouth can be a great way to increase a movie’s popularity. A small local movie theater released a movie. On the first day, only 5 people saw the movie. They all loved it, and each told at least 5 more people to go see the movie. The second day of the movie’s release, many of the people who had been told to see the movie went to the theater. Each day, each person who viewed the movie told approximately 5 other people to go to the theater. The table below shows the number of people who viewed the movie in its first 4 days out.