Questions: For the function involving the variables listed, use your intuition or additional research, if necessary, to complete parts (a) through (c) below. (day of year, average high temperature) over a 2-year period for a town a. Describe an appropriate domain and range for the function. Choose the appropriate domain for the function below. A. The domain is all daily average high temperatures over 2 years or -20°F to 120°F. B. The domain is all days over a 2-year period or 1 to 365, where day 1 is July 1st. C. The domain is all days over a 2-year period or 1 to 730, where day 1 is July 1st. D. The domain is all daily average high temperatures over 2 years or 50°F to 90°F. Choose the appropriate range for the function below. A. The range is all days over a 2-year period or 1 to 365, where day 1 is July 1st. B. The range is all daily average high temperatures over 2 years or 50°F to 90°F. C. The range is all daily average high temperatures over 2 years or -20°F to 120°F. D. The range is all days over a 2-year period or 1 to 730, where day 1 is July 1st. b. Make a rough sketch of a graph of the function. Choose the correct graph below. Let t represent time in days and F represent temperature in degrees Fahrenheit. c. Briefly discuss the validity of the graph as a model of the true function. Choose the correct answer below. A. The graph is a valid model of the function because time in days fluctuates as temperature in degrees Fahrenheit increases. B. The validity of the graph as a model of the true function can never be known. C. The graph is not a valid model of the true function. D. The graph is a valid model of the function because temperature in degrees Fahrenheit fluctuates as time in days go through seasons.

For the function involving the variables listed, use your intuition or additional research, if necessary, to complete parts (a) through (c) below. (day of year, average high temperature) over a 2-year period for a town a. Describe an appropriate domain and range for the function. Choose the appropriate domain for the function below. A. The domain is all daily average high temperatures over 2 years or -20°F to 120°F. B. The domain is all days over a 2-year period or 1 to 365, where day 1 is July 1st. C. The domain is all days over a 2-year period or 1 to 730, where day 1 is July 1st. D. The domain is all daily average high temperatures over 2 years or 50°F to 90°F. Choose the appropriate range for the function below. A. The range is all days over a 2-year period or 1 to 365, where day 1 is July 1st. B. The range is all daily average high temperatures over 2 years or 50°F to 90°F. C. The range is all daily average high temperatures over 2 years or -20°F to 120°F. D. The range is all days over a 2-year period or 1 to 730, where day 1 is July 1st. b. Make a rough sketch of a graph of the function. Choose the correct graph below. Let t represent time in days and F represent temperature in degrees Fahrenheit. c. Briefly discuss the validity of the graph as a model of the true function. Choose the correct answer below. A. The graph is a valid model of the function because time in days fluctuates as temperature in degrees Fahrenheit increases. B. The validity of the graph as a model of the true function can never be known. C. The graph is not a valid model of the true function. D. The graph is a valid model of the function because temperature in degrees Fahrenheit fluctuates as time in days go through seasons.
Transcript text: For the function involving the variables listed, use your intuition or additional research, if necessary, to complete parts (a) through (c) below. (day of year, average high temperature) over a 2-year period for a town a. Describe an appropriate domain and range for the function. Choose the appropriate domain for the function below. A. The domain is all daily average high temperatures over 2 years or $-20^{\circ} \mathrm{F}$ to $120^{\circ} \mathrm{F}$. B. The domain is all days over a 2-year period or 1 to 365 , where day 1 is July 1 st. C. The domain is all days over a 2 -year period or 1 to 730 , where day 1 is July 1 st. D. The domain is all daily average high temperatures over 2 years or $50^{\circ} \mathrm{F}$ to $90^{\circ} \mathrm{F}$. Choose the appropriate range for the function below. A. The range is all days over a 2 -year period or 1 to 365 , where day 1 is July 1 st. B. The range is all daily average high temperatures over 2 years or $50^{\circ} \mathrm{F}$ to $90^{\circ} \mathrm{F}$. C. The range is all daily average high temperatures over 2 years or $-20^{\circ} \mathrm{F}$ to $120^{\circ} \mathrm{F}$. D. The range is all days over a 2 -year period or 1 to 730 , where day 1 is July 1st. b. Make a rough sketch of a graph of the function. Choose the correct graph below. Let $t$ represent time in days and F represent temperature in degrees Fahrenheit. c. Briefly discuss the validity of the graph as a model of the true function. Choose the correct answer below. A. The graph is a valid model of the function because time in days fluctuates as temperature in degrees Fahrenheit increases. B. The validity of the graph as a model of the true function can never be known. C. The graph is not a valid model of the true function. D. The graph is a valid model of the function because temperature in degrees Fahrenheit fluctuates as time in days go through seasons.
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Solution

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Solution Steps

Step 1: Determine the domain

The domain represents the possible inputs for the function, which in this case are the days over a 2-year period. Since there are approximately 365 days in a year, a 2-year period has approximately 730 days. We'll start counting from July 1st of the first year.

Step 2: Determine the range

The range represents the possible outputs for the function, which in this case is the average high temperature. Reasonable average high temperatures throughout a year, even over two years, would fall between -20°F and 120°F, covering both extreme cold and heat.

Step 3: Choose the correct graph and discuss its validity.

The most appropriate graph would be B. It demonstrates a cyclic pattern representing temperature fluctuations over the two-year period along the x-axis, and also shows average high temperatures along the y-axis. This oscillatory nature of temperature aligns with how seasons change throughout the year, with peaks representing warmer months and troughs representing cooler months. Graph C also shows cyclic change. However, the changes along the t-axis (x-axis) are far too frequent to indicate temperature changes, and also we cannot see temperatures below 50 Fahrenheit in this graph. Thus, it is not a good model of temperature changes. Graph A and D do not exhibit an oscillatory pattern. The axes are also mislabeled in graph A.

The graph is a valid model because it represents cyclical changes and the axes are labeled correctly. It's important to note that this is a simplified model. Real-world temperature fluctuations aren't perfectly periodic and can be influenced by numerous factors.

Final Answer

Domain: All days over a 2-year period or 1 to 730, where day 1 is July 1st. Range: All daily average high temperatures over 2 years or -20°F to 120°F. Graph: B Validity: D. The graph is a valid model of the function because temperature in degrees Fahrenheit fluctuates as time in days go through seasons.

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