Arithmetic and Graphing: Solutions

Here are some solutions to the Exercises to accompany the section Arithmetic and Graphing.  

1. Complete the analysis of the function f(x) = -(x - 1)/2, by completing the following table:

   Algebraic	Numerical				Geometric
   	English	Values			English	Number Line
   x	Take the input	-1	0	1	Take the line
   x - 1	Subtract 1
   (x - 1)/2	Multiply by 1/2
   -(x - 1)/2	Multiply by -1

   Solution

   Algebraic	Numerical				Geometric
   	English	Values			English	Number Line
   x	Take the input	-1	0	1	Take the line
   x - 1	Subtract 1	-2	-1	0	Shift left 1
   (x - 1)/2	Multiply by 1/2	-1	-1/2	0	Shrink by 2
   -(x - 1)/2	Multiply by -1	1	1/2	0	Flip

   Back to Exercises.

2. Analyze each of the following functions, algebraically, numerically, and geometrically, as in the previous Exercise:

   1. g(x) = 2(x + 3)

      Solution

      Algebraic	Numerical				Geometric
      	English	Values			English	Number Line
      x	Take the input	-1	0	1	Take the line
      x + 3	Add 3	2	3	4	Shift right 3
      2(x + 3)	Multiply by 2	4	6	8	Stretch by 2

       

   2. h(x) = 2x + 6

      Solution

      Algebraic	Numerical				Geometric
      	English	Values			English	Number Line
      x	Take the input	-1	0	1	Take the line
      2x	Multiply by 2	-2	0	2	Stretch by 2
      2x + 6	Add 6	4	6	8	Shift right 6

       

   3. k(x) = -(x - 2)

      Solution

      Algebraic	Numerical				Geometric
      	English	Values			English	Number Line
      x	Take the input	-1	0	1	Take the line
      x - 2	Subtract 2	-3	-2	-1	Shift left 2
      -(x - 2)	Multiply by -1	3	2	1	Flip

       

   4. r(x) = -x + 2

      Solution

      Algebraic	Numerical				Geometric
      	English	Values			English	Number Line
      x	Take the input	-1	0	1	Take the line
      -x	Multiply by -1	1	0	-1	Flip
      -x + 2	Add 2	3	2	1	Shift right 2

   Back to Exercises.

3. Looking at the results of the previous Exercise:

   1. Compare g and h algebraically, numerically, and geometrically.

      Solution

      Algebraically, g(x) = 2(x + 3) = 2x + 6 = h(x), by the distributive law.  Numerically, they take the same values in the end, although the intermediate values differ.  Geometrically, g shifts then scales, while h reverses the order (scale then shift).  The order matters, in that we have to shift more after a stretch than before to obtain the same overall effect; that is because the shift factor is "stretched", too.  This gives a new, geometric way of viewing the distributive law, for positive numbers.
      

   2. Compare k and r algebraically, numerically, and geometrically.

      Solution

      Algebraically, k(x) = -(x - 2) = -x + 2 = r(x), by the distributive law.  Numerically, they take the same values in the end, although the intermediate values differ.  Geometrically, k shifts then flips, while h reverses the order (flip then shift).  The order matters, in that we have to shift in the opposite direction after a flip than before to obtain the same overall effect.  This gives another geometric way of viewing the distributive law, for negative signs.

   Back to Exercises.

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