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1. 6-1 Multiple Representations of Functions Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

2. Warm Up Create a table of values for each function. 1. y = 2x – 3 2. f(x) = 4x – x2

3. Objectives Translate between the various representations of functions. Solve problems by using the various representations of functions.

4. Example : Recreation Application Janet is rowing across an 80-meter-wide river at a rate of 3 meters per second. Create a table, an equation, and a graph of the distance that Janet has remaining before she reaches the other side. When will Janet reach the shore?

5. Example 2 Continued Step 1 Create a table. Let t be the time in seconds and d be Janet’s distance, in meters, from reaching the shore. Janet begins at a distance of 80 meters, and the distance decreases by 3 meters each second so we know it is linear.

6. Distance is equal to minus 80 3 meters per second. Example 2 Continued Step 2 Write an equation. d = 80 – 3t

7. 2 2 2 –80 t = = 26 3 3 3 –3 t-intercept: 26 Janet will reach the shore after seconds. 26 Example 2 Continued Step 3 Find the intercepts and graph the equation. d-intercept:80 Solve for t when d = 0 d = 80 – 3t 0 = 80 – 3t

8. Check It Out! Example 2 The table shows the height, in feet, of an arrow in relation to its horizontal distance from the archer. Create a graph, an equation, and a verbal description to represent the height of the arrow with relation to its horizontal distance from the archer.

9. Check It Out! Example 2 Continued Step 1 Use the table to check finite differences. First differences 53.25 30.75 8.25 –14.3 –37 Second differences –22.5 –22.5 –22.55 –22.7 Since second differences are close to constant, use the quadratic regression feature to write an equation.

10. 100 0 0 500 Check It Out! Example 2 Continued Step 2 Write an equation. Use the quadratic regression feature. –0.002x2 + 0.86x + 6.52 Step 3 Graph the equation. The archer shoots the arrow from a height of 6.52 ft. It travels horizontally about 220 ft to its peak height of about 99 ft, and then it lands on the ground about 500 ft away.

11. Population in US from 1790 to 1890 Use a graphing calculator to determine the exponential regression equation

12. How fast is the population growing? 33% each decade

13. Use the model to predict a population value for 1845. Recall that our data started in 1790 and goes by decades. How could we check to see if this is reasonable?

14. Population in US from 1790 to 1890 Use a graphing calculator to determine the exponential regression equation

15. We can look at residuals as one indicator of how well the model performs over time Residuals are the difference between the data value and the corresponding output value from the model If your data is in L1 and L2 then press and select RESID -0.19, -0.12, 0.01, 0.07, 0.27, 0.36, 1.01, 1.99, 0.83, -1.46, -5.47

16. -0.19, -0.12, 0.01, 0.07, 0.27, 0.36, 1.01, 1.99, 0.83, -1.46, -5.47 What does the sign of the residual reveal? Whether the data value is greater or less than the corresponding output value from the model Based on residuals, the model is not as good for x-values greater than 8 because the model begins to grow too quickly

17. How well does the model fit the data? r2 = .9979

18. Use the model to predict a population value for 1990. Recall that our data started in 1790 and goes by decades. How much confidence do you have in this prediction? Actual Population in 1990 was approximately 250 million

19. Ticket Out A hotel manager knows that the number of rooms that guests will rent depends on the price. The hotel’s revenue depends on both the price and the number of rooms rented. The table shows the hotel’s average nightly revenue based on room price. Use a graphing calculator to find the price a manager should charge in order to maximize his revenue.

20. Assignment: Worksheet Practice B

21. Ticket Out Answers The data do not appear to be linear, so check finite differences.

22. Ticket Out Answers Revenues 21,000 22,400, 23,400 24,000 First differences 1,400 1,000 600 Second differences 400 400 Because the second differences are constant, a quadratic model is appropriate. Use a graphing calculator to perform a quadratic regression on the data.

23. Ticket Out Answers The equation y = –2x2 + 440x models the data, and the graph appears to fit. Use the TRACE or MAXIMUM feature to identify the maximum revenue yield. The maximum occurs when the hotel manager charges $110 per room.

24. Day 2DiscussionFilm

25. Answers to Practice B

26. If you were asked to solve a problem, would you rather be given a verbal description, a table, a graph, or an equation? Why? Does the question you are asked effect your choice?

27. Given the following table, create a graph. Is this a function? What type of function? How do you know?

28. Come up with an equation to model this data. How confident are you in the model? Why? Is the model good for all data points? How do we check?

29. Talk to your neighbor and come up with a situation that might be modeled by this data

30. Check It Out! Example 3 Bartolo opened a new sporting goods business and has recorded his business sales each week. To break even, Bartolo needs to sell $48,000 worth of merchandise in a week. Assuming the sales trend continues, use a graph and an equation to find the number of weeks before Bartolo breaks even.

31. r1 = = = 1.1 r4 = = ≈ 1.1 r2 = = = 1.1 r3 = = = 1.1 36,603 27,500 30,250 33,275 y4 y3 y1 y2 27,500 25,000 33,275 30,250 y1 y0 y2 y3 Check It Out! Example 3 Continued The data appears to be exponential, so check for a common ratio. Because there is a common ratio, an exponential model is appropriate. Use a graphing calculator to perform an exponential regression on the data.

32. Check It Out! Example 3 Continued The equation y = 22,727.15(1.1)x models the data and the graph appears to fit. Use the TRACE feature to identify the value of x that corresponds to 48,000 for y. He will break even after 8 weeks.

34. Assignment: Practice C