Transformational Graphing: Practice Exercises

Here are various 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.

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.

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.

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
   2. g(x) = x/(1 + x²)
   3. h(x) = |x|(1 - x²)
   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.

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.
   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.

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

   1.  
   2. g(x) = -(2 - x)² + 1
   3. h(x) = 1 - (x - 2)²
   4.  

   Determine the pairs of these functions which are actually equal, and explain why this is so.

   Solution.

Go to Graphing and Mathematical Models .

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