Transformational Graphing: Solutions

Here are some solutions to the Exercises to accompany the section Transformational Graphing.  Some Exercises are designed to be done with a partner and to be graded by the partner.

Transformational Graphing

  1. Consider the function, f(x) = -(x - 1)2/3 - 2.
    1. Decompose this into a series of arithmetic operations with the "core" function, g(x) = x2, in the middle.
    2. Convert your list from a) into a series of algebraic expressions, starting with x2, that differ by one arithmetic operation at each step.
    3. Create a table with a column corresponding to each expression from b).
    4. Translate each algebraic step into the corresponding geometric and numerical effects.
    5. Calculate the corresponding input and output values at each step, starting from the points (-1, 1), (0, 0), and (1, 1).
    6. Using the fact that the graph of g is a parabola opening upwards:

      plot the points you calculated in e) sketch the graph of f.

    Solution
    1. This decomposes as, take the input and:
      • subtract 1,
      • square this,
      • multiply by -1,
      • divide by 3, and
      • subtract 2.
    2. This converts to the sequence of expressions:
      • x2
      • (x - 1)2
      • -(x - 1)2
      • -(x - 1)2/3
      • -(x - 1)2/3 - 2
    3.  We can analyze this by the following table:
      Corresponding
      Algebraic formula      		 x2 (x - 1)2   -(x - 1)2 -(x - 1)2/3 -(x - 1)2/3 - 2
      Geometric Effect Take the graph of
      the square function      		        
      Numerical Effect     Apply the
      square function      		      
      Numerical 
      Results      		 Inputs Outputs
      -1   1      
      0   0      
      1   1      
    4. We can translate each algebraic operation into the corresponding geometric and numerical effects:
      Corresponding
      Algebraic formula      		 x2 (x - 1)2   -(x - 1)2 -(x - 1)2/3 -(x - 1)2/3 - 2
      Geometric Effect Take the graph of
      the square function      		 Shift right 1 Flip vertically Shrink vertically
      by 3      		 Shift down 2
      Numerical Effect   Add 1 Apply the
      square function      		 Multiply by -1 Divide by 3 Subtract 2
      Numerical 
      Results      		 Inputs Outputs
      -1   1      
      0   0      
      1   1      
    5. Computing the corresponding input/output values:
      Corresponding
      Algebraic formula      		 x2 (x - 1)2   -(x - 1)2 -(x - 1)2/3 -(x - 1)2/3 - 2
      Geometric Effect Take the graph of
      the square function      		 Shift right 1 Flip vertically Shrink vertically
      by 3      		 Shift down 2
      Numerical Effect   Add 1 Apply the
      square function      		 Multiply by -1 Divide by 3 Subtract 2
      Numerical 
      Results      		 Inputs Outputs
      -1 0 1 -1 -1/3 -7/3
      0 1 0 0 0 -2
      1 2 1 -1 -1/3 -7/3
      To summarize, we have the following table of points:
      x f(x)
      0
      1
      2      		 -7/3
      -2
      -7/3

      that correspond to the original points on the square graph. 

    6. When we shift right, the vertex moves from (0, 0) to (1, 0), but it still opens upwards.  When we flip vertically, it opens downwards.  Shrinking vertically makes the graph increase less quickly (which will make the graph look wider), and shifting down moves the vertex down to (1, -2).  This looks like:

    Back to Exercises.

  2. Consider the function h given by the following verbal description:
    • Add 4,
    • divide the result by 2,
    • take the square root,
    • multiply by -1, and
    • add 3.
    1. Given an algebraic formula for h. 
    2. Use the transformational graphing technique, that you practiced in the previous Exercise, to create a table and sketch of h.
    3. Use XFunctions to create a plot each step in your table to check your work:

    Solution
    1. Applying each operation in order gives  .
    2. Analysis of this function gives:
      Corresponding
      Algebraic formula      		  
      Geometric Effect Take the graph of
      the square root function      		 Stretch horizontally
      by 2      		 Shift left
      4 units      		 Flip vertically Shift up 3
      Numerical Effect   Multiply by 2 Subtract 4 Apply the square
      root function      		 Multiply by -1 Add 3
      Numerical 
      Results      		 Inputs Outputs
      0 0 -4 0 0 3
      1 2 -2 1 -1 2
      4 8 4 2 -2 1
      To summarize, we have the following table of points:
      x f(x)
      -4
      -2
      4      		 3
      2
      1

      that correspond to the original points on the square root graph. 

      The final graph has its vertex at (-4, 3) and goes down and right, curving upwards through the points (-6, 2) and (4, 1):
    3. The series of Examples in the following applet shows the graph at each step:

    Back to Exercises.

  3. Give a verbal description of a function, similar to the previous Exercise, using the absolute value as the "core" function to create a similar practice Exercise.  Repeat the steps of the previous Exercise using your example. 
    Solution
    Use the Multigraph Utility in the following applet to plot each of the steps in your table.  Make sure to:
    • set the limits on each axis to include all the points of each graph,
    • check that each graph is related to the next by the geometric transformation that you expected, and
    • verify that each graph goes through the indicated points.

    Note: You should repeat this Exercise as often as necessary, with different "core" functions, until you have mastered our transformational graphing technique.

    Back to Exercises

Applications to Algebra

  1. Use XFunctions to graph each of the following functions and:
    1. Decide whether the function is even or odd.
    2. Depending on what you decide, use algebra to show that either f(-x) = f(x) or f(-x) = -f(x).
    1. f(x) = x2 - x4
      Solution
      This graph:

      is clearly symmetric with respect to horizontal flips, so we will try to show that f(-x) = f(x).  This is straightforward, using rules of exponents: f(-x) = (-x)2 - (-x)4 = (-1)2(x)2 - (-1)2(x)4 = x2 - x4 = f(x).  This proves that this is an even function.
    2. g(x) = x/(1 + x2)
      Solution
      This graph:

      is clearly symmetric with respect to a half-rotation around the origin, so we will try to show that g(-x) = -g(x).  As before, we use rules of exponents: g(-x) = (-x)/(1 + (-x)2) = -(x/(1 + x2)) = -g(x).  This proves that this is an odd function.
    3. h(x) = |x|(1 - x2)
      Solution
      This graph:

      is clearly symmetric with respect to the vertical axis, so we will try to show that h(-x) = h(x).  Now we use the multiplication property of the absolute value function (and rules of exponents): h(-x) = |-x|(1 - (-x)2) = |-1| |x|(1 - x2) = |x|(1 - x2) = g(x).  This shows that this function is even .
    4. .  Note: You will need to use the "cubert" function in XFunctions; to do the algebra, however, you will probably want to write the cube root as a fractional exponent and use rules of exponents.
      Solution
      This graph:

      is clearly symmetric with respect to 180° rotations around the origin.  Using rules of exponents, we have: k(-x) = ((-x) - (-x)3)1/3 = (-x - (-1)3x3)1/3 = ((-1)3x - (-1)3x3)1/3 = ((-1)3(x - x3)1/3 = (-1)3/3(x - x3)1/3 = (-1)(x - x3)1/3 = -(x - x3)1/3 = k(x), which verifies that this is an odd function.

    Back to Exercises.

  2. Work with your partner to create more practice Exercises.
    1. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make a table of values at two different inputs for his/her function, and show it to you.  Use the point-slope formula to guess your partner's function; create a transformational graphing table to verify your formula.
      Solution
      Your partner should correct your work.
    2. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make define a new function with this formula in XFunctions:

      Use the point-slope formula to guess your partner's function; create a transformational graphing table to verify your formula.

      Solution
      Look at the function definition in XFunctions, or ask your partner.  

    Back to Exercises.

  3. Use our transformational graphing method to graph each of the following functions.

    1.  
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		  
      Geometric Effect Take the graph of the
      square root function      		 Shift left 4 Flip horizontally Shrink horizontally by 2
      Numerical Effect   Subtract 4 Negate Divide by 2 Apply the square
      root function
      Numerical 
      Results      		 Inputs Outputs
      0 -4 4 2 0
      1 -3 3 3/2 1
      4 0 0 0 2
      The final graph has its vertex at (2, 0) and goes up and left, curving downwards through the points (3/2, 1) and (0, 2):
    2. g(x) = -(2 - x)2 + 1
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		 x2 (2 + x)2 (2 - x)2   -(2 - x)2 -(2 - x)2 + 1
      Geometric Effect Take the graph of
      the square function      		 Shift left 2 units Flip horizontally Flip vertically Shift up 1
      Numerical Effect   Subtract 2 Negate Apply the
      square function      		 Negate Add 1
      Numerical 
      Results      		 Inputs Outputs
      -1 -3 3 1 -1 0
      0 -2 2 0 0 1
      1 -1 1 1 -1 0
      The final graph has its vertex at (2, 1) and looks like a parabola that opens downwards through the points (3, 0) and (1, 0):
    3. h(x) = 1 - (x - 2)2
      Solution
      This can be analyzed as follows:
      Corresponding
      Algebraic formula      		 x2 (x - 2)2   -(2 - x)2 1 - (2 - x)2
      Geometric Effect Take the graph of the
      square function      		 Shift right 2 units Flip vertically Shift up 1
      Numerical Effect   Add 2 Apply the
      square function      		 Negate Add 1
      Numerical 
      Results      		 Inputs Outputs
      -1 1 1 -1 0
      0 2 0 0 1
      1 3 1 -1 0
      The final graph goes through the same points,  (3, 0) and (1, 0), with vertex at (2, 1), and the graph is just the same as in part b).  That is because, algebraically, g(x) = -(2 - x)2 + 1 = 1 - (2 - x)2 = 1 - (-(2 - x))2 =  1 - (2 - x)2 = h(x).  We first used the commutative law of addition to switch the order of the terms, and the equivalence of subtraction and addition of a negative.  We then factored out a -1 from inside the square, and then used the fact that the square function is even.
    4.  
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		  
      Geometric Effect Take the graph of
      the square root function      		 Shift left 1 unit Flip horizontally Stretch horizontally
      by 2      		 Stretch vertically by 2
      Numerical Effect   Subtract 1 Negate Multiply by 2 Apply the square
      root function      		 Multiply by 2
      Numerical 
      Results      		 Inputs Outputs
      0 -1 1 2 0 0
      1 0 0 0 1 2
      4 3 -3 -6 2 4
      The final graph has its vertex at (2, 0) and goes up and left, curving downwards through the points (0, 2) and (-6, 4):
      We can see that this is the same as f, just with different points plotted.  This is because, algebraically, if we factor out a 4:

       

      Note: Since roots are just fractional exponents, we can use the rule of exponents to "distribute" the root across the product.

       

    Back to Exercises.

Go to Graphing and Mathematical Models.

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