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Functional Shifts

Objective: To be able to describe the effects of a function when the equation is algebraically manipulated.

Materials: GSP

Time: 1-2 days

Level of Difficulty: Medium

Procedure:

Investigation 1: Absolute value

NOTE: When graphing the absolute value function on Algebra Xpresser you must use the abbreviation abs for the absolute value.

1) Using Algebra Xpresser, graph the function y = Describe the general shape of the function.

2) Now graph y = + 1 .Be sure to keep the original function on the screen. (Be sure to use "or" in between the equations. Describe what happened.

3) Now graph the following functions on the same axes:

y = + 2

y = + 5

y = + 8

Sketch a graph of the five functions. Make a conjecture based on your findings. Make a generalization for y = + a, where a is a real number.

4) Now change the values of a in the previous investigation to their additive inverses. Describe the changes in the graph. Now make additional generalizations for y = + a.

5) Now start a new graph with y = . Predict what the graph of

y = will look like. Now graph the function y = . Did the graph resemble what you predicted?

6) Now graph the following on the same set of axes:

y =

y =

y =

7) Make a sketch of the functions from #6. Make a conjecture that explains how the graph is affected when a real number is added inside the absolute value sign.

Investigation 2: Square roots

NOTE: When graphing the square root on Algebra Xpresser you must use the abbreviation sqrt for the square root.

1) Using Algebra Xpresser, graph the function y = Describe the general shape of the function.

2) Now graph y = + 1 .Be sure to keep the original function on the screen. (Be sure to use "or" in between the equations. Describe what happened.

3) Now graph the following functions on the same axes:

y = + 2

y = + 5

y = + 8

Sketch a graph of the five functions. Make a conjecture based on your findings. Make a generalization for y = + a, where a is a real number.

4) Now change the values of a in the previous investigation to their additive inverses. Describe the changes in the graph. Now make additional generalizations for y = + a.

5) Now start a new graph with y = . Predict what the graph of

y = will look like. Now graph the function y = . Did the graph resemble what you predicted?

6) Now graph the following on the same set of axes:

y =

y =

y =

7) Make a sketch of the functions from #6. Make a conjecture that explains how the graph is affected when a real number is added inside the square root sign.