7.1C Check Up Solution

Compare your answers

1. There is a linear relationship between the Celsius and Fahrenheit temperature scales. The following table shows some equivalent temperatures.
   
   

   1. Use the table to sketch a graph of the relation relating the Celsius and Fahrenheit scales.
      
      
      
      

   2. 
      

      1. Estimate the slope and y-intercept of your graph.
         
         Estimates will vary.  Slope is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#FF0000¨»«mn»9«/mn»«mn»5«/mn»«/mfrac»«/math»or 1.8 and y-intercept is 32.
         
         

      2. Determine an equation for the graph, in slope-intercept form.
         
         Use the slope formula and two points on the graph to determine the slope. 
         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtext mathcolor=¨#FF0000¨»m§#160;=«/mtext»«mfrac mathcolor=¨#FF0000¨»«mrow»«mn»50«/mn»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»40«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mn»10«/mn»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»40«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mtext mathcolor=¨#FF0000¨»=«/mtext»«mfrac mathcolor=¨#FF0000¨»«mn»9«/mn»«mn»5«/mn»«/mfrac»«mtext mathcolor=¨#FF0000¨»=1.8«/mtext»«/math»
          
         
         Substitute a point into «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mi mathcolor=¨#FF0000¨»C«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«/math»  to determine the b -value.
         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtext mathcolor=¨#FF0000¨»Substitute§#160;(10,§#160;50).«/mtext»«mspace linebreak=¨newline¨/»«mn mathcolor=¨#FF0000¨»50«/mn»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mfenced mathcolor=¨#FF0000¨»«mn»10«/mn»«/mfenced»«mo mathcolor=¨#FF0000¨»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«mspace linebreak=¨newline¨/»«mn mathcolor=¨#FF0000¨»50«/mn»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»18«/mn»«mo mathcolor=¨#FF0000¨»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«mspace linebreak=¨newline¨/»«mn mathcolor=¨#FF0000¨»32«/mn»«mo mathcolor=¨#FF0000¨»=«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mi mathcolor=¨#FF0000¨»C«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«/math»
         
         An equation relating the Celsius and Fahrenheit scales is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mi mathcolor=¨#FF0000¨»C«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«/math».
         
         

      3. How did your estimated slope and y-intercept values compare to the calculated values?
         
         Responses will vary.
         
         

   3. 
      

      1. What does the slope represent in this scenario?
         
         The slope represents how much the Fahrenheit scale changes for each change of 1°C. For every increase of 1°C, the Fahrenheit change is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mo mathcolor=¨#FF0000¨»§#176;«/mo»«/math».
         
         

      2. What does the y-intercept represent in this scenario?
         
         The y-intercept represents the Fahrenheit temperature at 0°C.
         This means 0°C = 32°F.
         
         

   4. 1. What is 140ºC in Fahrenheit?
         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mfenced mathcolor=¨#FF0000¨»«mn»140«/mn»«/mfenced»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»252«/mn»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»258«/mn»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mn mathcolor=¨#FF0000¨»140«/mn»«mo mathcolor=¨#FF0000¨»§#176;«/mo»«mtext mathcolor=¨#FF0000¨»C§#160;equals§#160;258§#176;F«/mtext»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/math»
         
         

      2. What is –80ºF in Celsius?
         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#FF0000¨»-«/mo»«mn mathcolor=¨#FF0000¨»80«/mn»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mi mathcolor=¨#FF0000¨»C«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«mspace linebreak=¨newline¨/»«mo mathcolor=¨#FF0000¨»-«/mo»«mn mathcolor=¨#FF0000¨»112«/mn»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mi mathcolor=¨#FF0000¨»C«/mi»«mspace linebreak=¨newline¨/»«mo mathcolor=¨#FF0000¨»-«/mo»«mn mathcolor=¨#FF0000¨»62«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»2«/mn»«mo mathcolor=¨#FF0000¨»§#8773;«/mo»«mi mathcolor=¨#FF0000¨»C«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo mathcolor=¨#FF0000¨»-«/mo»«mn mathcolor=¨#FF0000¨»80«/mn»«mo mathcolor=¨#FF0000¨»§#176;«/mo»«mtext mathcolor=¨#FF0000¨»F§#160;approximately§#160;equals§#160;-62.2§#176;C«/mtext»«/math»
         
         
         

   5. Theoretically, the coldest possible temperature occurs at approximately -273.15°C and is sometimes referred to as “absolute zero”. Use this information to state the domain and range of the temperature conversion relation.
      
      The minimum Celsius temperature corresponds to the minimum Fahrenheit temperature.
      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mn mathcolor=¨#FF0000¨»1«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»8«/mn»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo»-«/mo»«mn»273«/mn»«mo».«/mo»«mn»15«/mn»«/mrow»«/mfenced»«mo mathcolor=¨#FF0000¨»+«/mo»«mn mathcolor=¨#FF0000¨»32«/mn»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»F«/mi»«mo mathcolor=¨#FF0000¨»=«/mo»«mo mathcolor=¨#FF0000¨»-«/mo»«mn mathcolor=¨#FF0000¨»459«/mn»«mo mathcolor=¨#FF0000¨».«/mo»«mn mathcolor=¨#FF0000¨»67«/mn»«mo mathcolor=¨#FF0000¨»§#176;«/mo»«mtext mathcolor=¨#FF0000¨»F«/mtext»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»Domain«/mi»«mo mathcolor=¨#FF0000¨»:«/mo»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mfenced mathcolor=¨#FF0000¨ open=¨[¨ close=¨]¨»«mrow»«mo»-«/mo»«mn»273«/mn»«mo».«/mo»«mn»15«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«mspace linebreak=¨newline¨/»«mi mathcolor=¨#FF0000¨»Rangle«/mi»«mo mathcolor=¨#FF0000¨»:«/mo»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mfenced mathcolor=¨#FF0000¨ open=¨[¨ close=¨]¨»«mrow»«mo»-«/mo»«mn»459«/mn»«mo».«/mo»«mn»67«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/math»