Classwork Exercises
The vertices π΄π΄(0,0), π΅π΅(1,0), πΆπΆ(1,1), and π·π·(0,1) of a unit square can be represented by the complex numbers π΄π΄= 0, π΅π΅= 1, πΆπΆ= 1 +ππ, and π·π·=ππ.
1. Let πΏπΏ1(π§π§) =βπ§π§.
a. Calculate π΄π΄β²=πΏπΏ1(π΄π΄), π΅π΅β²=πΏπΏ1(π΅π΅), πΆπΆβ²=πΏπΏ1(πΆπΆ), and π·π·β²=πΏπΏ1(π·π·). Plot these four points on the axes.
b. Describe the geometric effect of the linear transformation πΏπΏ1(π§π§) =βπ§π§ on the square π΄π΄π΅π΅πΆπΆπ·π·.
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2. Let πΏπΏ2(π§π§) = 2π§π§.
a. Calculate π΄π΄β²=πΏπΏ2(π΄π΄), π΅π΅β²=πΏπΏ2(π΅π΅), πΆπΆβ²=πΏπΏ2(πΆπΆ), and π·π·β²=πΏπΏ2(π·π·). Plot these four points on the axes.
b. Describe the geometric effect of the linear transformation πΏπΏ2(π§π§) = 2π§π§ on the square π΄π΄π΅π΅πΆπΆπ·π·.
3. Let πΏπΏ3(π§π§) =πππ§π§.
a. Calculate π΄π΄β²=πΏπΏ3(π΄π΄), π΅π΅β²=πΏπΏ3(π΅π΅), πΆπΆβ²=πΏπΏ3(πΆπΆ), and π·π·β²=πΏπΏ3(π·π·). Plot these four points on the axes.
b. Describe the geometric effect of the linear transformation πΏπΏ3(π§π§) =πππ§π§ on the square π΄π΄π΅π΅πΆπΆπ·π·.
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4. Let πΏπΏ4(π§π§) = (2ππ)π§π§.
a. Calculate π΄π΄β²=πΏπΏ4(π΄π΄), π΅π΅β²=πΏπΏ4(π΅π΅), πΆπΆβ²=πΏπΏ4(πΆπΆ), and π·π·β²=πΏπΏ4(π·π·). Plot these four points on the axes.
b. Describe the geometric effect of the linear transformation πΏπΏ4(π§π§) = (2ππ)π§π§ on the square π΄π΄π΅π΅πΆπΆπ·π·.
5. Explain how transformations πΏπΏ2, πΏπΏ3, and πΏπΏ4 are related.
6. We will continue to use the unit square π΄π΄π΅π΅πΆπΆπ·π· with π΄π΄= 0, π΅π΅= 1, πΆπΆ= 1 +ππ, π·π·=ππ for this exercise.
a. What is the geometric effect of the transformation πΏπΏ(π§π§) = 5π§π§ on the unit square?
b. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππ)π§π§ on the unit square?
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c. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππ2)π§π§ on the unit square?
d. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππ3)π§π§ on the unit square?
e. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππ4)π§π§ on the unit square?
f. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππ5)π§π§ on the unit square?
g. What is the geometric effect of the transformation πΏπΏ(π§π§) = (5ππππ)π§π§ on the unit square, for some integer ππ β₯0?
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Exploratory Challenge
Your group has been assigned either to the 1-team, 2-team, 3-team, or 4-team. Each team will answer the questions below for the transformation that corresponds to their team number:
πΏπΏ1(π§π§) = (3 + 4ππ)π§π§ πΏπΏ2(π§π§) = (β3 + 4ππ)π§π§ πΏπΏ3(π§π§) = (β3β4ππ)π§π§ πΏπΏ4(π§π§) = (3β4ππ)π§π§.
The unit square π΄π΄π΅π΅πΆπΆπ·π· with π΄π΄= 0, π΅π΅= 1, πΆπΆ= 1 +ππ, π·π·=ππ is shown below. Apply your transformation to the vertices of the square π΄π΄π΅π΅πΆπΆπ·π·, and plot the transformed points π΄π΄β², π΅π΅β², πΆπΆβ², and π·π·β² on the same coordinate axes.
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For the 1-team:
a. Why is π΅π΅β²= 3 + 4ππ?
b. What is the argument of 3 + 4ππ?
c. What is the modulus of 3 + 4ππ?
For the 2-team:
a. Why is π΅π΅β²=β3 + 4ππ?
b. What is the argument of β3 + 4ππ?
c. What is the modulus of β3 + 4ππ?
For the 3-team:
a. Why is π΅π΅β²=β3β4ππ?
b. What is the argument of β3β4ππ?
c. What is the modulus of β3β4ππ?
For the 4-team:
a. Why is π΅π΅β²= 3β4ππ?
b. What is the argument of 3β4ππ?
c. What is the modulus of 3β4ππ?
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All groups should also answer the following:
a. Describe the amount the square has been rotated counterclockwise.
b. What is the dilation factor of the square? Explain how you know.
c. What is the geometric effect of your transformation πΏπΏ1, πΏπΏ2, πΏπΏ3, or πΏπΏ4 on the unit square π΄π΄π΅π΅πΆπΆπ·π·?
d. Make a conjecture: What do you expect to be the geometric effect of the transformation πΏπΏ(π§π§) = (2 +ππ)π§π§ on the unit square π΄π΄π΅π΅πΆπΆπ·π·?
e. Test your conjecture with the unit square on the axes below.
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Problem Set
1. Find the modulus and argument for each of the following complex numbers.
a. π§π§1=οΏ½ 2 3 + 1 2 ππ b. π§π§2= 2 + 2β3ππ c. π§π§3=β3 + 5ππ d. π§π§4=β2β2ππ e. π§π§5= 4β4ππ f. π§π§6= 3β6ππ
2. For parts (a)β(c), determine the geometric effect of the specified transformation.
a. πΏπΏ(π§π§) =β3π§π§ b. πΏπΏ(π§π§) =β100π§π§ c. πΏπΏ(π§π§) =β13π§π§
d. Describe the geometric effect of the transformation πΏπΏ(π§π§) =πππ§π§ for any negative real number ππ.
3. For parts (a)β(c), determine the geometric effect of the specified transformation.
a. πΏπΏ(π§π§) = (β3ππ)π§π§ b. πΏπΏ(π§π§) = (β100ππ)π§π§ c. πΏπΏ(π§π§) =οΏ½β13πποΏ½ π§π§
d. Describe the geometric effect of the transformation πΏπΏ(π§π§) = (ππππ)π§π§ for any negative real number ππ.
4. Suppose that we have two linear transformations, πΏπΏ1(π§π§) = 3π§π§ and πΏπΏ2(π§π§) = (5ππ)π§π§.
a. What is the geometric effect of first performing transformation πΏπΏ1 and then performing transformation πΏπΏ2? b. What is the geometric effect of first performing transformation πΏπΏ2 and then performing transformation πΏπΏ1? c. Are your answers to parts (a) and (b) the same or different? Explain how you know.
5. Suppose that we have two linear transformations, πΏπΏ1(π§π§) = (4 + 3ππ)π§π§ and πΏπΏ2(π§π§) =βπ§π§. What is the geometric effect of first performing transformation πΏπΏ1 and then performing transformation πΏπΏ2?
6. Suppose that we have two linear transformations, πΏπΏ1(π§π§) = (3β4ππ)π§π§ and πΏπΏ2(π§π§) =βπ§π§. What is the geometric effect of first performing transformation πΏπΏ1 and then performing transformation πΏπΏ2?
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7. Explain the geometric effect of the linear transformation πΏπΏ(π§π§) = (ππ β ππππ)π§π§, where ππ and ππ are positive real numbers.
8. In Geometry, we learned the special angles of a right triangle whose hypotenuse is 1 unit. The figures are shown above. Describe the geometric effect of the following transformations.
a. πΏπΏ1(π§π§) =οΏ½οΏ½23+12πποΏ½ π§π§ b. πΏπΏ2(π§π§) =οΏ½2 + 2β3πποΏ½π§π§ c. πΏπΏ3(π§π§) =οΏ½οΏ½22+οΏ½22πποΏ½ π§π§ d. πΏπΏ4(π§π§) = (4 + 4ππ)π§π§
9. Recall that a function πΏπΏ is a linear transformation if all π§π§ and π€π€ in the domain of πΏπΏ and all constants ππ meet the following two conditions:
i. πΏπΏ(π§π§+π€π€) =πΏπΏ(π§π§) + πΏπΏ(π€π€) ii. πΏπΏ(πππ§π§) =πππΏπΏ(π§π§)
Show that the following functions meet the definition of a linear transformation.
a. πΏπΏ1(π§π§) = 4π§π§ b. πΏπΏ2(π§π§) =πππ§π§ c. πΏπΏ3(π§π§) = (4 +ππ)π§π§
10. The vertices π΄π΄(0, 0), π΅π΅(1, 0), πΆπΆ(1, 1), π·π·(0, 1) of a unit square can be represented by the complex numbers π΄π΄= 0, π΅π΅= 1, πΆπΆ= 1 +ππ, π·π·=ππ. We learned that multiplication of those complex numbers by ππ rotates the unit square by 90Β° counterclockwise. What do you need to multiply by so that the unit square will be rotated by 90Β° clockwise?
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