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Comparing models

Comparing models. Lines, quadratics, and exponential equations. This table of values shows the depreciation of the value of a car from year to year, where x is the number of years after it was purchased, which was in 2000 . Calculate the rate of change in value from 2000 to 2003 .

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Comparing models

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  1. Comparing models Lines, quadratics, and exponential equations

  2. This table of values shows the depreciation of the value of a car from year to year, where x is the number of years after it was purchased, which was in 2000. • Calculate the rate of change in value from 2000 to 2003. • What units should be used todescribe the rate of change? • Describe the rate of change. Minds on

  3. Rate of change is calculated by finding: Rate of Change = Change in y Change in x • aka Change in y / Change in x • So the units work the same way! • UnitsforRate of Change = Unitsfory / Unitsforx Units for Rate of Change

  4. In a distance vs. time graph, the units for the rate of change could be… • In a table showing earnings over time, the units for the rate of change could be… • In measuring the class average against the number of students added to a class, the units would be… Examples

  5. I can distinguish between linear, quadratic, and exponential models • I can compare pairs of relations Learning Goals

  6. Linear Models Characteristics of graphs, equations, rates of change, and first differences

  7. A linear model represents quantities that increase or decrease by a constant amount over equal intervals • In a table of values, the first differences are equal • The graph is a straight line • The equation can be written in the form y = m x + b, where m = slope, and b = y-intercept • The rate of change is constant Linear Models

  8. 500-W power setting 1000-W power setting How do these situations compare? A cup of coffee is reheated in a microwave. There are 2 power settings. The temperature of the coffee, C degrees Celsius, after t seconds in the microwave can be modeled by the above.

  9. This table shows the median age of Canada’s population from 1975 to 2000. • A) Determine the equation of the line of best fit • B) Predict the median age of Canada’s population in 2020. • With the graphing calculator • Without the graphing calculator Fitting a Linear Model to Data

  10. y = 0.38x – 723.3 http://www.meta-calculator.com/online/ Fitting a Linear Model to Data

  11. Pg. 293 # 1-3, 7, 10, 14 • Homework Quiz on MONDAY! Homework

  12. Graphical Model Investigation Complete in pairs using the graphing calculators

  13. Quadratic Models How are they different from linear models?

  14. I can describe quadratic models in terms of their: • Visual graphs • Equations • Rates of change • First and second differences Learning Goals

  15. http://www.thirteen.org/get-the-math/the-challenges/math-in-basketball/introduction/181/http://www.thirteen.org/get-the-math/the-challenges/math-in-basketball/introduction/181/ Math in Basketball

  16. In a table of values, the second differences are equal, and not zero • The graph is a curve called a parabola • The equation can be written in the form ax2+bx+c =0where a is not zero. WHY??? • The rate of change is always changing  either from increasing to decreasing or from decreasing to increasing Characteristics of Quadratic Models

  17. EXAMPLE 1 Stopping Distances The graph shows the stopping distance (in metres) and speed of a car (in kilometres per hour) (a) Describe the relationship between stopping distance and speed As speed increases, the stopping distance also increases (b) Use the graph to estimate the stopping distance at (i) 50 km / h At 50 km/h, stopping distance is approximately 10 metres (ii) 100 km / h At 100 km/h, stopping distance is approximately 40 metres

  18. EXAMPLE 1 Stopping Distances (c)  How many times further is the stopping distance at 100 km/h compared to the stopping distance at 50 km/h? (d)Consider the rate of change of stopping distance with respect to speed. What are the appropriate units for this rate of change? The stopping distance at 100 km/h is 4 times further than the stopping distance at 50 km/h The appropriate units are metres per kilometre per hour

  19. EXAMPLE 1 Stopping Distances (e)  Is the rate of change of stopping distance with respect to speed increasing, constant or decreasing? Explain. The rate of change is increasing  Graph is curving up and is gradually getting steeper

  20. EXAMPLE 2 Analysing a Free Fall The table shows distance and time data for the Drop Zone ride at Canada’s Wonderland. (a) Does the time column show equal time intervals? Yes  Each time interval is going up by 0.2 seconds + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2

  21. EXAMPLE 2 • Analysing a Free Fall • (b) Calculate the 1st Differences and record it in the table. • Does this model a linear relationship? Explain. • No •  1st Differences are not constant • (ii) Do the 1st Differences imply an increasing, constant or decreasing rate of change of distance with respect to time? • Since 1st differences are becoming more positive, the rate of change is increasing + 0.2 0.2 – 0.0 = + 0.2 + 0.2 0.8 – 0.2 = + 0.6 + 0.2 1.8 – 0.8 = + 1.0 + 0.2 + 1.4 + 0.2 + 1.8 + 0.2 + 2.2 + 0.2 + 2.6 + 0.2 + 3.0 + 3.4 + 0.2 + 0.2 + 3.8 + 4.2 + 0.2 + 0.2 + 4.6 + 0.2 + 5.0

  22. EXAMPLE 2 Analysing a Free Fall (c) Calculate and record the 2nd Differences. Do the 2nd Differences imply a quadratic model? Explain. Yes  2nd Differences are constant + 0.2 0.6 – 0.2 = + 0.4 + 0.6 1.0 – 0.6 = + 0.4 + 1.0 1.4 – 1.0 = + 0.4 + 1.4 + 0.4 + 1.8 + 0.4 + 2.2 + 0.4 + 2.6 + 0.4 + 3.0 + 0.4 + 3.4 + 0.4 + 3.8 + 0.4 + 4.2 + 0.4 + 4.6 + 0.4 + 5.0 PREVIOUS

  23. EXAMPLE 2 Analysing a Free Fall (d) Create a scatterplot with time on the horizontal axis and distance on the vertical axis. Time vs. Distance Distance (metres) Time (seconds)

  24. EXAMPLE 2 Analysing a Free Fall (e)  Does the graph imply a linear or non-linear relation? NON-LINEAR  Points form a smooth curve (f) Compare the table and the graph. Does the rate of change of distance with respect to time appear to be increasing, constant or decreasing. INCREASING - curve on graph is going up and becoming steeper Time vs. Distance Distance (metres) Time (seconds)

  25. How can I tell if I’m looking at quadratic data? • First differences are NOT equal • Second differences are equal • Graph looks like a curve – specifically parabolic • Equation has a squared term in it: usually written as x2 Consolidate

  26. I can describe quadratic models in terms of their: • Visual graphs • Equations • Rates of change • First and second differences Homework: Pg. 303 # 1-3, 5-8, 10 Learning Goals

  27. I can describe exponential models in terms of their: • Visual graphs • Equations • Rates of change • First and second differences Learning Goals

  28. EXPONENTIAL MODELS KEY CONCEPTS For all exponentialmodels, the first and second differences are always non-constant The ratios can be calculated by DIVIDING each y-value by the y-value that comes BEFORE it. When RATIOS are CONSTANT, the relation is exponential. In an exponential relation, there is a constant percent increase over equal intervals

  29. EXAMPLE 1 Bacteria Growth (b) Use the graph to estimate the number of bacteria after each time period (i) 20 minutes ________ (ii) 40 minutes ________ (iii) 60 minutes ________ (c) Calculate the ratios Divide the number of bacteria after 40 min by the number after 20 min Divide the number of bacteria after 60 min by the number after 40 min 20 40 80

  30. EXPONENTIAL MODELS EXAMPLE 1 Bacteria Growth Divide the number of bacteria after 40 min by the number after 20 min Divide the number of bacteria after 60 min by the number after 40 min (d) What happens to the number of bacteria every 20 minutes? The number of bacteria doubles

  31. EXPONENTIAL MODELS EXAMPLE 1 Bacteria Growth (e) Consider the rate of change of number of bacteria with respect to time. What are suitable units for this rate of change? (f) Is the rate of change of number of bacteria with respect to time increasing, constant or decreasing. Explain. INCREASING  Curve is going upwards  Curve is getting steeper The units for rate of change is bacteria per minute

  32. EXAMPLE 2 • Smoke Detectors • Americium-241 (Am-241) is a manufactured element. It is a silvery radioactive metal, which is used in smoke detectors. Household smoke detectors contain about 200 micrograms (g) of Am-241. The amount of Am-241 present in the detector decreases or decays over time. The table shows the mass of Am-241 remaining, in micrograms, over 1000 years. • Does the Years column show equal time intervals? • Yes •  Going up by 100 years + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100

  33. EXPONENTIAL MODELS EXAMPLE 2 Smoke Detectors (b)  Calculate the 1st Differences and 2nd Differences. Is the relationship linear, quadratic or neither? Explain. • 1st Differences • Not constant • Not Linear! + 100 170 - 200 = – 30 + 100 145 - 170 = – 25 + 100 124 – 145 = – 21 + 100 – 19 + 100 – 15 + 100 – 14 + 100 – 11 + 100 – 10 + 100 – 8 + 100 – 7 PREV

  34. EXPONENTIAL MODELS EXAMPLE 2 Smoke Detectors (b)  Calculate the 1st Differences and 2nd Differences. Is the relationship linear, quadratic or neither? Explain. • 1st Differences • Not constant • Not Linear! • 2nd Differences • Not constant • Not Quadratic! – 25 – (– 30) = – 25 + 30 = + 5 + 100 – 30 + 100 + 5 – 25 + 4 – 21 – (– 25) = – 21 + 25 = + 4 + 100 – 21 + 2 + 100 – 19 + 4 + 100 – 15 + 100 + 1 – 14 + 100 + 3 – 11 + 1 + 100 – 10 – 19 – (– 21) = – 19 + 21 = + 2 + 2 + 100 – 8 + 1 + 100 – 7

  35. EXAMPLE 2 Smoke Detectors (c) Calculate the ratios. Divide the second mass by the first, the third mass by the second, etc. How do you know that the relation is exponential? • 1st Differences • Not constant • Not Linear! • 2nd Differences • Not constant • Not Quadratic! • Ratios • Constant!!! • EXPONENTIAL! + 0.85 + 0.85 Divide successive y values! + 0.85 + 0.85 170 / 200 = + 0.85 + 0.85 + 0.85 + 0.85 145 / 170 = + 0.85 + 0.85 + 0.85 124 / 145 = + 0.85 + 0.85

  36. EXPONENTIAL MODELS EXAMPLE 2 Smoke Detectors (d) Draw a graph with Years on the horizontal axis and Mass remaining on the vertical axis. (e) How do you know that this graph is: (i) Not linear? Not a straight line (ii) Not quadratic? Unable to tell (would need to look beyond given data) Mass (g) Time (Years)

  37. EXPONENTIAL MODELS EXAMPLE 2 Smoke Detectors (d) Draw a graph with Years on the horizontal axis and Mass remaining on the vertical axis. • (f) Compare the table and graph. Does the rate of change of mass remaining with respect to years appear to be constant, increasing or decreasing? Explain. • Look at 1st Differences • They are increasing • BUT... • Graph is decreasing •  Curve is becoming less steep Mass (g) Time (Years)

  38. EXPONENTIAL MODELS EXAMPLE 2 Smoke Detectors (g) What are the suitable units for the rate of change of mass remaining with respect to years? The units for the rate of change of mass remaining with respect to years is micrograms per year

  39. I can describe exponential models in terms of their: • Visual graphs • Equations • Rates of change • First and second differences Learning Goals

  40. Exponential Models With technology

  41. EXPONENTIAL REGRESSION EXAMPLE Draw Cards Sandor drew cards from a standard deck of 52 playing cards until he drew a heart. He repeated this experiment many times. Each time, Sandor recorded the number of cards he drew before drawing a heart. For example, if he drew a heart on the first draw, then he drew zero cards before he drew a heart. This happened 50 times. (a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship between frequency and number of cards appear to be linear, quadratic or exponential. Explain. • 1st Differences • Not constant • Not Linear! 38 – 50 = – 12 28 – 38 = – 10 21 – 28 = – 7 – 5 – 4 – 3

  42. EXPONENTIAL REGRESSION EXAMPLE Draw Cards Sandor drew cards from a standard deck of 52 playing cards until he drew a heart. He repeated this experiment many times. Each time, Sandor recorded the number of cards he drew before drawing a heart. For example, if he drew a heart on the first draw, then he drew zero cards before he drew a heart. This happened 50 times. (a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship between frequency and number of cards appear to be linear, quadratic or exponential. Explain. – 10 – (– 12) – 10 + 12 = + 2 • 1st Differences • Not constant • Not Linear! • 2nd Differences • Not constant • Not Quadratic! – 12 + 2 – 10 – 7 – (– 10) – 7 + 10 = + 3 + 3 – 7 + 2 – 5 + 1 – 4 + 1 – 3 – 5 – (– 7) – 5 + 7 = + 2

  43. EXPONENTIAL REGRESSION EXAMPLE Draw Cards Sandor drew cards from a standard deck of 52 playing cards until he drew a heart. He repeated this experiment many times. Each time, Sandor recorded the number of cards he drew before drawing a heart. For example, if he drew a heart on the first draw, then he drew zero cards before he drew a heart. This happened 50 times. (a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship between frequency and number of cards appear to be linear, quadratic or exponential. Explain. 38 / 50 = + 0.76 • 1st Differences • Not constant • Not Linear! • 2nd Differences • Not constant • Not Quadratic! • Ratios • Relatively CONSTANT • EXPONENTIAL! + 0.76 28 / 38 = + 0.74 + 0.74 + 0.75 21 / 28 = + 0.75 + 0.76 + 0.75 + 0.75

  44. EXPONENTIAL REGRESSION EXAMPLE Draw Cards EXPONENTIAL REGRESSION 1. You need to set-up your calculator so it can perform exponential regression. Press 2nd 0 (zero) x-1 Use the DOWN cursor key until you reach Diagnostic On. Once you reach this command, press ENTER and ENTER 2. Clear the data table by pressing STAT 4:ClrList 2ndand 1 then “,” 2ndand 2 (* your screen should look like the one on the bottom right) ENTER ClrList L1, L2

  45. EXPONENTIAL REGRESSION EXAMPLE Draw Cards EXPONENTIAL REGRESSION 3. Enter the data into the lists by pressing STAT and 1:Edit. Enter the “# OF CARDS” data in L1and “FREQUENCY” data in L2. 4. Create a scatter plot by pressing 2nd, Y=, 1 and ENTER. Make sure that the cursor is on “On” when you press Enter. By default, your Xlist should be L1 and Ylist should be L2. 5. To display your graph, press ZOOM and 9.

  46. EXPONENTIAL REGRESSION EXAMPLE Draw Cards EXPONENTIAL REGRESSION 3. Enter the data into the lists by pressing STAT and 1:Edit. Enter the “# OF CARDS” data in L1and “FREQUENCY” data in L2. 4. Create a scatter plot by pressing 2nd, Y=, 1 and ENTER. Make sure that the cursor is on “On” when you press Enter. By default, your Xlist should be L1 and Ylist should be L2. 5. To display your graph, press ZOOM and 9. Sketch the scatterplot below. Label all axes Frequency # of Cards

  47. EXPONENTIAL REGRESSION EXAMPLE Draw Cards REGRESSION ANALYSIS and EXPONENTIAL EQUATION OF BEST FIT 1. To perform regression analysis, press STAT Move the right cursor over to CALC Press 0:ExpReg (this activates the exponential regression function) 2. You must tell the calculator which data to perform the quadratic regression on. We do this by entering the lists from our data table. You do this by pressing 2nd, 1 (this will tell it to pick L1) then “,” (comma key) 2nd, 2 (this will tell it to pick L2) then “,” (comma key)

  48. EXPONENTIAL REGRESSION EXAMPLE Draw Cards REGRESSION ANALYSIS and EXPONENTIAL EQUATION OF BEST FIT 3. We will store the information for our line of best fit in a variable called Y1. To do this, press: VARS Rightcursor key to Y-VARS 1:Function 1:Y1 Your screen should look like this: 4. Press ENTER to execute. Fill the information that is given to you: (a) Write the exponential equation of best fit _________________________ (b) Press GRAPH to view the scatterplot with the exponential curve of best fit. Draw the curve on your sketch in #5 above. ExpReg y = a*b^x a = b = r2 = r = 50.028 0.751 0.999 0.999 y = 50.028(0.751)x ExpReg L1, L2,Y1

  49. EXPONENTIAL REGRESSION EXAMPLE Draw Cards 4. Press ENTER to execute. Fill the information that is given to you: (a) Write the exponential equation of best fit _________________________ (b) Press GRAPH to view the scatterplot with the exponential curve of best fit. Draw the curve on your sketch in #5 above. ExpReg y = a*b^x a = b = r2 = r = Frequency 50.028 0.751 0.999 0.999 # of Cards y = 50.028(0.751)x

  50. EXPONENTIAL REGRESSION EXAMPLE Draw Cards (c)  Use the equation from 4(a) to answer to predict the frequency of drawing eight (8) cards before drawing a heart Substitute x = 8 into the equation y = 50.028(0.751)x y = 50.028(0.751)8 y = 50.028(0.1012) y = 5 The frequency of drawing 8 cards before drawing a heart is 5 4. Press ENTER to execute. Fill the information that is given to you: (a) Write the exponential equation of best fit _________________________ (b) Press GRAPH to view the scatterplot with the exponential curve of best fit. Draw the curve on your sketch in #5 above. ExpReg y = a*b^x a = b = r2 = r = 50.028 0.751 0.999 0.999 y = 50.028(0.751)x

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