643 6.4 Translations of the Graphs of the Sine and Cosine Functions (a) Predict the average annual temperature. (b) Plot the average monthly temperature over a two-year period, letting x = 1 correspond to January of the first year. (c) Determine a function of the form ƒ1x2 = acos3b1x - d24 + c, where a, b, c, and d are constants, that models the data. (d) Graph ƒ together with the data on the same coordinate axes. How well does ƒ model the given data? (e) Use the sine regression capability of a graphing calculator to find the equation of a sine curve that fits these data (over a two-year interval). (Modeling) Monthly Temperatures A set of temperature data (in °F) is given in the tables for a particular location. (Data from www.weatherbase.com) (a) Plot the data over a two-year interval. (b) Use sine regression to determine a model for the two-year interval. Let x = 1 represent January of the first year. (c) Graph the equation from part (b) together with the given data on the same coordinate axes. 65. Average Monthly Temperature, Buenos Aires, Argentina Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 77.2 74.7 70.5 63.9 57.7 52.2 51.6 54.9 57.6 63.9 69.1 73.8 66. Average High Temperature, Buenos Aires, Argentina Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 86.7 83.7 79.5 72.9 66.2 60.1 58.8 63.1 66.0 72.5 77.5 82.6 64. Average Monthly Temperature The average monthly temperature (in °F) in Phoenix, Arizona, is shown in the table. (Modeling) Fractional Part of the Moon Illuminated The data in the tables give the fractional part of the moon that is illuminated during the month indicated. (Data from http://aa.usno.navy.mil) (a) Plot the data for the month. (b) Use sine regression to determine a model for the data. (c) Graph the equation from part (b) together with the given data on the same coordinate axes. 67. January 2019 Day 1 2 3 4 5 6 7 8 9 10111213141516 Fraction 0.22 0.14 0.08 0.03 0.01 0.00 0.01 0.04 0.09 0.15 0.22 0.30 0.40 0.49 0.60 0.70 Day 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Fraction 0.79 0.87 0.94 0.98 1.00 0.98 0.94 0.87 0.78 0.68 0.57 0.47 0.36 0.27 0.19 68. November 2019 Day 1 2 3 4 5 6 7 8 91011121314 Fraction 0.20 0.29 0.38 0.48 0.58 0.67 0.75 0.83 0.89 0.94 0.98 1.00 0.99 0.97 Day 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Fraction 0.93 0.86 0.78 0.68 0.58 0.46 0.35 0.25 0.15 0.08 0.03 0.00 0.00 0.03 0.08 0.14 Month °F Month °F Jan 56 July 95 Feb 60 Aug 94 Mar 65 Sept 88 Apr 73 Oct 77 May 82 Nov 64 June 91 Dec 55 Data from National Climatic Data Center.
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